I'm going to assume that you know to clear all the rights in advance before you bring somebody else's character into your own story, and just go into the actual storytelling side (something I actually have some experience with, as my first professional book contract was licensed fiction).
So, what I was told right after I signed the contract to write Diablo: Demonsbane was that the rule with licensed fiction is that you have to return the characters and the setting in the condition in which you found them. So, I could have Archangel Tyrael appear and interact with my characters, but I couldn't kill him off, maim him, etc. I could have characters visit Tristram (I actually don't - I picked an undeveloped part of the map called Engstieg and dropped a bunch of pre-conquest Anglo-Saxons there), but I couldn't blow the place up.
You're looking at similar restrictions. You can bring somebody else's character (with their creator's permission, of course) into your story, have them interact with your own characters, and integrate them into the storyline. But, unless their creator wants to develop them in some manner in your story, you must return them as you found them (so, they have to be the same person at the end of the crossover as they were at the beginning).
What I was provided for Demonsbane was called a "bible" - literally, a guide to the setting and established characters that I would be allowed to use that told me everything I needed to know (and, as a result, I knew there would be a Diablo 3 back in 2000, because they put in a section of "what happens after Diablo 2, and please don't make reference to this unless you absolutely have to"). Ideally, you want to get one of these from the creator of each character you use so that you have a full rundown of who that character is, how they see the world, what they want, and how they are likely to react under stress, etc.
And then you use that material to build your story.