My figures are in an air-conditioned environment. Heat may accelerate the process but I don't think it's fully responsible for the "stickiness". I tend to agree with those that think it's a chemical reaction between the plastics.
Yep. Plasticizers are volatile chemicals and as heat increases, they "migrate" at a faster rate: It's that new plastic aroma we love. Flexible plastics are inherently less stable than rigid plastics, and that softness comes from the plasticizers. As plasticizers "evaporate", the plastic shrinks and becomes less flexible. Unfortunately, they can cause surface tackiness and can react with other plastics they contact, softening/melting them. You can clean the surface, but more will keep migrating from the substrate to the surface. It eventually slows down, but I don't think there are any rules about how long it takes.
The reaction can do a lot of damage to hard plastics: Judging by the reaction I noticed in my bases, I speculate that, if left in place, they'd eventually melt through the base, leaving a gummy goopy hole that doesn't harden (I've seen this happen before, and it's ugly).
Part of this is our own doing-- To meet the tight production/release schedules, manufacturers can't keep stuff around in their factories to give the stuff time to properly cure, since that could take weeks, or months, or longer. We want our stuff ASAP, which usually means almost as soon as it's unloaded from the boats: It doesn't get the cure time that it might if it were sitting on a retailer's shelves. We unbox it and immediately put it in the stand, or put the blaster in its hand (Flexible hands are another thing to watch out for).
Unfortunately, I don't think you can really do anything about it except let it do its thing and cure, monitoring it, and taking precautions to ensure that other stuff doesn't get damaged through contact.
Jim