(Don't mind me, just spouting off stuff I noticed after going back and replaying a few FF games.)
Not to the extent of, say, Cid being in every game, but it sure does happen a lot:

Final Fantasy V has Exdeath kill everyone in the world with the void near the end (albeit temporarily), but said world has both a village of Moogles and a village of Salem Wolf Mans. Still pretty mild since everyone comes back, but still just the beginning.

Final Fantasy VI has a tribe of Moogles living in Narshe, but it's implied only Mog actually survives to the end.

Final Fantasy VII kills off most of its Fire-Wolf-Lion things in its backstory. (To its credit there's still apparently enough survivors for Red XIII to have babbies in the ending.)

Final Fantasy VIII is actually a step up, what with the Moombas merely beeing enslaved rather than murdered. (Although I could just be forgetting something; VIII's stream-of-consciousness approach to plot points makes it easy for me to forget stuff.)

Final Fantasy IX snaps right on back and goes on to slaughter the Burmecians. Twice. And then reduces Freya, one of the few survivors, to basically a background character and does nothing satisfying with her own subplots. Credit for explicitly leaving a few more alive than usual, I suppose.

Final Fantasy X has the Ronso get killed by a totally bullshit boss figh- err, Seymour. And he went and taunted Kimahri about being the "last Ronso" (even though he wasn't). What a dick.
(Also, ChipCheezum was right; Kimahri is totally Worf if Worf was an anthropomorphic blue cat man.)

Final Fantasy XII gets stealthy about this; it populates its world with a multitude of furries... and is a prequel to Final Fantasy Tactics, a game where only humans exist and has a vague mass-extinction-causing catastrophe in its backstory.

Dayumn. I mean, I get the appeal*; I wish I could launch all furries into the Void too whenever an influential person in the fandom espouses a bizarre right-wing/libertarian conspiracy theory or insists "fursecution" is a real prejudice like homophobia or racism or does something equally shitty, but this still strikes me as a bit much.
Kinda wonder if this'll ever happen again, now that new Final Fantasy games try to go for "realistic" art styles.

*I also get that the real reason this happens is the writers need redshirts writ large so they use non-humans on the incorrect assumption that people will be less emotionally invested.