I picked up the Steel Striker shield and the Worn Dragonplate helm and have used them in a couple of missions. (Sarven Brawl, Yakstraction, The Great Yak Stampete) and have had to re-equip them each time as the equip state isn’t saving.
They aren’t restricted on these levels. It’s not game breaking, since they can be re-equipped each time.
Are you going forward to missions you haven’t played before, or back to older missions that you had played with older gear? It should remember your gear for new levels, but it should keep the older gear you had when you last played a level you’d already started. Is that what’s happening, perhaps?