I recently moved barns as well! The previous barn owner never made sure his horses had water and fed them cattle hay. My advice is to check on your horse as often as possible after the move to make sure they are adapting well and not having any issues, whether that be with other horses, or changes in surroundings. Keep an eye out for anxiety.
I just recently moved barns, and everyone is so nice and friendly! You have to make an effort to give it your all and be nice. Always ask questions about things you don't know about as things may be different