When I sent my girl away for training (out of state), I only packed the essentials. As others say, I don’t send things I wasn’t okay with not getting back or being torn up. The trainer supplied grooming supplies and all tack (my mare wasn’t under saddle yet so she didn’t have anything fitted to her).
I sent along bell boots, fly masks, fly boots, a tail bag, the rest of her feed (barn fed the same thing she was already on), and some leave-in detangler I wanted finished off. Also sent some feed products I wasn’t going to use for the barn to have (Probos and something else). If my horse needed anything else, the trainer let me know and I had it shipped there (bridle, boots, winter blankets, etc.) or I took it myself when I visited. Anything that isn’t needed got dumped back in the trunk; that trunk wasn’t get next to her stall or anything.
My trunk(s) at home are another matter. Where I boarded, I had a tack shed so more storage space and I crammed it full of anything I could possibly need. For a daily-use trunk, I kept gloves (1 for riding, 2 for lunging/handling, 3 for winter, 4 all the spares), hair nets (which I never used), helmet, boots, spurs, crop/whip, treats, small human first-aid kit, chapstick, horse boots (turnout, bells, and open fronts), sprays (fly, daily detangler, & hydrating spray), lunge line, spare halter, rope halter, spare lead, half pad, saddle pad and small saddle bag. I had an entire zippered grooming tote that I sometimes kept in there, but usually it was in my car. There is a seperate area for storing riding tack (hook for bridle + breast collar & rack for saddle + cinch + saddle pad).
Edit: Oh, can’t forget I keep a spare stud chain (or two) and a hoofpick in there as well.