Question about the Keeper's trait-
Can you carry a figure using this trait?

Here's a situation where it might come into question-

I'm fielding a team including Keeper and gravity feed Drax and facing a team including chase Thanos. I'm going first and wanting to get Drax within striking distance of Thanos before he draws in my two highest point characters (Drax not being one of them). In this instance I would want to utilize Keeper's trait in order to carry that Drax within a striking distance for next round. Is there anything preventing me from carrying Drax in this manner?

No, you're placing him, not moving him. You can only carry when you're moving (See Pip the Troll for more on this ruling)