Why did the Naugrim cross the Ered Luin in the first place?

[Beorn]

It came to pass during the second age of the captivity of Melkor that Dwarves came over the Blue Mountains of Ered Luin into Beleriand. Themselves they named Khazâd, but the Sindar called them Naugrim, the Stunted People, and Gonnhirrim, Masters of Stone. Far to the east were the most ancient dwellings of the Naugrim, but they had delved for themselves great halls and mansions, after the manner of their kind, in the eastern side of Ered Luin; and those cities were named in their own tongue Gabilgathol and Tumunzahar. To the north of the great height of Mount Dolmed was Gabilgathol, which the Elves interpreted in their tongue Belegost, that is Mickleburg; and southward was delved Tumunzahar, by the Elves named Nogrod, the Hollowbold. Greatest of all the mansions of the Dwarves was Khazad-dûm, the Dwarrowdelf, Hadhodrond in the Elvish tongue, that was afterwards in the days of its darkness called Moria; but it was far off in the Mountains of Mist beyond the wide leagues of Eriador, and to the Eldar came but as a name and rumour from the words of the Dwarves of the Blue Mountains.
From Nogrod and Belegost the Naugrim came forth into Beleriand; and the Elves were filled with amazement, for they had believed themselves to be the only living things in Middle-earth that spoke with words or wrought with hands, and that all others were but birds and beasts. But they could understand no word of the tongue of the Naugrim, which to their ears was cumbrous and unlovely; and few ever of the Eldar have achieved the mastery of it. But the Dwarves were swift to learn, and indeed were more willing to learn the Elventongue than to teach their own to those of alien race. Few of the Eldar went ever to Nogrod and Belegost, save Eöl of Nan Elmoth and Maeglin his son; but the Dwarves trafficked into Beleriand, and they made a great road that passed under the shoulders of Mount Dolmed and followed the course of the River Ascar, crossing Gelion at Sarn Athrad, the Ford of Stones, where battle after befell. Ever cool was the friendship between the Naugrim and the Eldar, thought much profit they had one of the other; but at that time those griefs that lay between them had not yet come to pass, and King Thingol welcomed them. But the Naugrim gave their friendship more readily to the Noldor in after days than to any others of Elves and Men, because of their love and reverence for Aulë; and the gems of the Noldor they praised above all other wealth. In the darkness of Arda already the Dwarves wrought great works, for even from the first days of their Fathers they had marvellous skill with metals and with stone; but in that ancient time iron and copper they loved to work, rather than silver or gold.

Source: Silmarillion, Chapter 10, 2nd & 3rd paragraphs

So, why did the Naugrim cross the Ered Luin in the first place?

 

[Aulë]

Well I would believe that it would be to search for new areas to mine. The Dwarves were in their prime back then, and the halls of Belegost, Nogrod and Khazad-dum would have been full.
They would have been looking for suitable places to build new dwellings and to extract iron and other metals.

I'm sure that there would have been a few 'adventurous' Dwarves who would have wanted to explore the 'unknown' and to discover new things.
The Longbeards were moving eastwards (to Erebor and the Iron Hills), so I see no reason why the Firebeards and the Broadbeams wouldn't travel westwards.

 

[Beleg]

The Dwarves were in their prime back then, and the halls of Belegost, Nogrod and Khazad-dum would have been full.

We cannot assuredly say that, certainly not about Moria which didn't reach it prime till the second age. And as for the mensions of Belegost and Nogrod, we are never supplied with any detail concerning them so we have to rely on conjucture while talking about them. But Ered Lindon is a pretty long chain of mountains and since the Dwarvish mines of Blue Mountains were there even in the third age, we cannot say that there was no space for more extraction.
Remember only the petty dwarves, the outcasts settled in Beleriand [or so we are told], and since they were the outlaws of their respective house it is just possible that they had thought Beleriand as an ideal place to dwell since we are indicated that Beleriand could have been rich in earthen minerals. [Take the valley of Tum-laden for examples and the caverns of Narog could also have been rich].
The other forays would mainly have been for exploration and for trade with the Sindar that dwelt there.

The Longbeards were moving eastwards (to Erebor and the Iron Hills), so I see no reason why the Firebeards and the Broadbeams wouldn't travel westwards.

These two settlements weren't settled, or reported to have been settled, way until into the third age so they don't come into contention. The travels of Longbeards in the Years of Trees aren't revealed to us but a more likely hypothesis is that they settled in Moria and along the Misty Mountains; not move eastwards.

 

[Inderjit S]

There were some Petty-Dwarves who were living in Beleriand. They claim to have lived there prior to the Eldar arriving in Beleriand. The accuracy of this statement is debatable, it may be that the Petty-Dwarves were attempting to assert their claim as the first settlers of Beleriand, and thus to claim some pieces of land as their own.

The Petty-dwarves. The Eldar did not at first recognize these as Incarnates, for they seldom caught sight of them in clear light. They only became aware of their existence indeed when they attacked the Eldar by stealth at night, or if they caught them alone in wild places. The Eldar therefore thought that they were a kind of cunning two-legged animals living in caves, and they called them Levain tad-dail, or simply Tad-dail, and they hunted them. But after the Eldar had made the acquaintance of the Naugrim, the Tad-dail were recognized as a variety of Dwarves and were left alone. There were then few of them surviving, and they were very wary, and too fearful to attack any Elf, unless their hiding-places were approached too nearly. The Sindar gave them the names Nogotheg 'Dwarflet', or Nogoth niben 'Petty Dwarf'.

Quendi and Eldar; HoME 11

The Dwarves were in a special position. They claimed to have known Beleriand before even the Eldar first came there; and there do appear to have been small groups dwelling furtively in the highlands west of Sirion from a very early date: they attacked and waylaid the Elves by stealth, and the Elves did not at first recognize them as Incarnates, but thought them to be some kind of cunning animal, and hunted them. By their own account they were fugitives, driven into the wilderness by their own kin further east, and later they were called the Noegyth Nibin or Petty-dwarves, for they had become smaller than the norm of their kind, and filled with hate for all other creatures. When the Elves met the powerful Dwarves of Nogrod and Belegost, in the eastern side of the Mountains, they recognized them as Incarnates, for they had skill in many crafts, and learned the Elvish speech readily for purposes of traffic. At first the Elves were in doubt concerning them, believing them to be related to Orcs and creatures of Morgoth; but when they found that, though proud and unfriendly, they could be trusted to keep any treaties that they made, and did not molest those who left them in peace, they traded with them and let them come and go as they would

Quendi and Eldar; HoME 11

The Dwarves were in their prime back then, and the halls of Belegost, Nogrod and Khazad-dum would have been full

Moria was pretty 'empty' (in comparison) at the end of the F.A. It only began to fill up after the S.A 40 and the influx of the Firebeard and Broadbeams.

40: Many Dwarves leaving their old cities in Ered Luin go to Moria and swell it's numbers

Tale of Years; LoTR

After the end of the First Age, the power and wealth of Khazad-dum was much increased

Appendix A; Durin's Folk

It's people 'began to dwindle' sometime after they closed their doors on Sauron, after attacking him from the rear when they aided Elrond, after the sacking of Ost-in-Edhil.

Firebeards and the Broadbeams wouldn't travel westwards.

Indeed. It strikes me odd that they took so much time to explore Beleriand, since their places of awakening were in the Ered Lindon.

In the Dwarvish traditions of the Third Age the names of the places where each of the Seven Ancestors had 'awakened' were remembered; but only two of them were known to Elves and Men of the West: the most westerly, the awakening place of the ancestors of the Firebeards and the Broadbeams; and that of the ancestor of the Longbeards, the eldest in making and awakening. The first had been in the north of the Ered Lindon, the great eastern wall of Beleriand, of which the Blue Mountains of the Second and later ages were the remnant;

Of Dwarves and Men; HoME 12

Unless they did explore some of Eastern Beleriand. They may have found it desolate, since the Sindar (Or their various sub-sectors) were said to prefer to live in woody areas, and Eastern Beleriand, esp. the Hills of Himring, and Estolad and land to the south of Estolad, was pretty tree-less. Ossiriand was a forest area, and some of Thargelion was held to contain forests too. So maybe the Dwarves went to Beleriand and found it to be desolate, at first? The Sindar, though a scattered peoples, mainly lived in three areas, the 'Mithrim', who lived around the lake Mithrim, in Hithlum, the Falthrim who lived in Western Beleriand (The Dwarves were said to hate the sea, anyway) and the Iathrim, who lived within Doriath. The Dwarves were said in Annals of Aman (HoME 10) to come in contact with the Sindar before the Nandor arrived in Beleriand. (Though there is a discrepancy in Quendi and Eldar to this statement, I think Tolkien may have forgotten about it

These two settlements weren't settled, or reported to have been settled, way until into the third age so they don't come into contention

This process began not in barter and trade, but in war; for the Longbeards had spread southward down the Vales of Anduin and had made their chief 'mansion' and stronghold at Moria; and also eastward to the Iron Hills, where the mines were their chief source of iron-ore. They regarded the Iron Hills, the Ered Mithrin, and the east dales of the Misty Mountains as their own land. But they were under attack from the Orks of Morgoth

Of Dwarves and Men (HoME 12)

Erebor wasn't settled until Thrain I, son of Nain, who was slain by the Balrog of Khazad-dum, found the Arkenstone and he prospered. But most of the Dwarves of Khazad-dum went to live in the Grey Mountains and so his son Thorin I later moved there because of it's natural wealth. But the Dragons that had been multiplying to the North of the Grey Mountains came and made war with the Dwarves and when Dain I, his great-grandson, was slain by a cold-drake (As well as his younger son, Gror) in T.A 2589, most of the Dwarves abandoned the Grey Mountains and went to reside in Erebor with Thror as the new King.