There are two challenge pack items that aren’t available to unlock in H3: the Jeager Covert sniper rifle from the Sapienza Master Sniper Challenge Pack and the Claw Hammer from the Plumber’s Apprentice pack (also in Sapienza).
H3 also doesn’t have Holiday Hoarders, the Hokkaido Snow Festival or the Mills Reverie escalation in Hawke’s Bay, so you can’t unlock the items associated with those either.
I haven’t transferred progress myself but it’s my understanding that Sniper Assassin mastery is reset back to zero, though you do keep the sniper rifle if you’ve unlocked it.
Escalations and Featured Contracts have a different unlock system but I think most of the unlocks (at least from the previous levels) are already available in H2. (Full list of unlocks is here.)
And, frankly, I have no idea how buggy the whole process is right now.
In case anyone cares, on PS4 I had all the trophies for H2 and they all popped in H3 immediately after the carryover. Three full minutes of trophies popping. I feel really accomplished!
The only trophies that didn’t pop automatically are the Sniper Assassin trophies. You have to do those again, but since you keep your challenges, a single SA run per map does the trick.
Since I played H3 before the carryover, I also deleted my local save files on PS4 after carrying over but before restarting H3. I’m not sure if that’s any help with the bugs, but I didn’t have any challenges that stayed completed without getting credit like some other players.