During the companion quests, followers tend to be replaced with one of the companions without warning. When this happens, the followers say "I'll head back home if you need me", and start walking away.
When I finish the quest and come back, do these followers still have all the items I gave them, or should I reload a previous save, unload all their equipment, and then go to Jorrvaskr?
They don't lose the equipment you give them, but they will revert to their old loadout. You will need to give or take a piece of armor or a weapon for them to equip the best things they have.
Also, watch out for invisible companion gear duplication, sometimes it happens after dismissing and rehiring a follower, leading to their decreased carry weight capacity. You may be able to pickpocket the duplicates after dismissing a follower, if they are unequipped, or, if you're on PC, remove them via the console.
When I finally purchased my first house in Whiterun my previous companion, Lydia, was there with her items still equipped.
