The only way that would work would be if the "Character Set" had the same name, but then you couldn't load them into the same scene. You could select the clip and export>clip. This will save the clip as something like "run.ma". In the new scene, create a character set for your new character with the same name. Then, you can import clips through the trax editor and they will be added to your visor. The clips will NOT import if there is even the slightest difference in you riggs.
The objects in your character sets are usually only the "controllers" that drive the bones in your skeleton. So, as long as those controllers haven't changed, you should be fine. However, seeing that the proportions of the new character will be different, this will mess up any IK translate information within the clip.
An easier way would be to use Maya's retargeting, but even that's a little tricky. When it comes to using animation clips on multiple riggs, this is where MotionBuilder shines over Maya. FBIK for Maya is almost there though.