Any collectibles should be stored or kept in a temperature controlled environment. 50F-78F is the general safe zone for most items. Putting your things in cases is not the biggest of your issues if their actual display space is in excess of 80F, and worse yet 90F-100F. I've worked with more than a few collectibles, including antique dolls, and early plastic dolls, and temperature is one of the biggest enemies to any collectible and will accelerate degradation of plastic. You have to worry about heat, light and moisture.
I store my collectibles out of light, in a temperature and humidity controlled environment. My antique Japanese dolls reside in glass cases. Plastic cases can leech out and degrade themselves, which can cause harm to collectibles. Glass or acrylic are preferable. Also avoid lots of varishing, proper lacquer is okay, like what is on my Japanese dolls cases, but varnish, especially with a humid environment, can leech out and yellow plastics, even when not in direct contact.