As you say, the Moriquendi just refers to all those Elves who never went to Aman. That's a pretty large and diverse group, and it's hard to make sweeping statements about the whole, but we can subdivide1
In the First Age, these are Thingol's people. Some of them likely left Middle-earth, but many others migrated into Eriador. A small number of these (chief among them Oropher, grandfather of Legolas) blended with the Silvan Elves who were already there:
Oropher had come among [the Silvan] with only a handful of Sindar, and they were soon merged with the Silvan Elves, adopting their language and taking names of Silvan form and style. This they did deliberately; for they (and other similar adventurers forgotten in the legends or only briefly named) came from Doriath after its ruin and had no desire to leave Middle-earth, nor to be merged with the other Sindar of Beleriand, dominated by the Noldorin Exiles for whom the folk of Doriath had no great love.
Unfinished Tales Part Two: "The Second Age" Chapter IV: "The History of Galadriel and Celeborn" Appendix B The Sindarin Princes of the Silvan Elves
Most of the rest became lesser nobility under the remaining Noldor in their kingdoms (Lothlórien, for instance)
This group is boring, because they never crossed the Misty Mountains, and chose to dwell in the vale of Anduin; so they never got to Beleriand, and are uninteresting for the purposes of this question.
Unfortunately there's not much known about what happened to these cousins of the Silvan. However, there's no indication that they need have perished at all; the land they lived in, Ossiriand, was called Lindon by the Noldor (emphasis mine):
In Ossiriand dwelt the Green-elves, in the protection of their rivers; for after Sirion Ulmo loved Gelion above all the waters of the western world. The woodcraft of the Elves of Ossiriand was such that a stranger might pass through their land from end to end and see none of them. They were clad in green in spring and summer, and the sound of their singing could be heard even across the waters of Gelion; wherefore the Noldor named that country Lindon, the land of music, and the mountains beyond they named Ered Lindon, for they first saw them from Ossiriand.
The Silmarillion III Quenta Silmarillion Chapter 14: "Of Beleriand and its Realms"
And we know that Lindon was one of the few regions of Beleriand to survive the War of Wrath; it's where Gil-Galad had his kingdom in the Second Age.
There's even less written about the Avari than about the Laiquendi, so we have to piece their fates together from some very limited information.
Some of them got picked up by the Laiquendi when they entered back into Ossiriand:
The Nandor had turned away, never seen the sea or even Osse, and had become virtually Avari. They had also picked up various Avari before they came back west to Ossiriand.
History of Middle-earth XI The War of the Jewels Part One: "The Grey Annals"
It's said later that some other Avari merged with some Eldar in other places as well; given the place names, they likely mingled with the Sindar and the Silvan Elves:
[T]he Lindarin elements in the western Avari were friendly to the Eldar, and willing to learn from them; and so close was the feeling of kinship between the remnants of the Sindar, the Nandor, and the Lindarin Avari, that later in Eriador and the Vale of Anduin they often became merged together.
History of Middle-earth XI The War of the Jewels Part Four: "Quendi and Eldar" Chapter C: "The Clan-names, with notes for names for divisions of the Eldar"
A later note suggests that few, if any, Avari even made it to Beleriand in the first place:
Those who had never made the journey to the West Shores were called 'the Refusers' (Avari). It is doubtful if any of the Avari ever reached Beleriand or were actually known to the Numenoreans.
History of Middle-earth XII The Peoples of Middle-earth Chapter X "Of Dwarves and Men"
But we're starting down a deep rabbit-hole of canonicity, so I'm going to stop here.
1 If you're less familiar with the varied and sundry Elvish sub-groups, you may find the answers to Major Stackings' Is there more than 1 species of elf in the Hobbit/LOTR saga? helpful (disclosure: one of them is mine)