I play year round in NY/NJ area. (yeah, we're a little crazy). I don't use my AZ Gloveman custom unless I'm on turf. I have older gloves that I use during the Winter and in bad weather.
My gamer, I'll condition it after the Fall leagues. Then, I usually leave it in the house with a ball and play catch, on the sofa, during the winter.
So no real storage process, other than the conditioning once a year.