This recently came up at my place of work. I only spent a couple days on this and my info could be off, but I'm fairly sure this is how it works.
FBX files convert models, smooth skins, joints, and controls to editable meshes, skin modifiers, dummys, and lines. It doesn't convert animation tools like constraints, utility maps, or expressions. This means you can transfer your model with the smooth skin intact from Maya to Max, but the controls won't work. To transfer animations, you'll need to bake your animations directly to the joints.
The following is a MELscript for baking translation and rotation animation to joints. To use it, select all the joints you're using in the smooth skin and run this script. The animation should play as it bakes to the joints. It might give you some warning messages, but they're just to inform you it's deleting some old keys that would interfere with the new keys. I tested it in Maya 8.0.
for ($bone in ls -sl)
{
bakeResults -t (playbackOptions -q -min + ":" + playbackOptions -q -max) -sm true -hi both -at t -at r $bone ;
}
To get it to bake scale data, type "-at s" near the end so it looks like "-at t -at r -at s $bone;"