Link the Animation Data.
- Select object without the keyframes first.
- Select the object with the keyframes to be copied second by holding shift and left clicking.
- Press ctrl+L
- Select "Animation Data"
This has in effect, copied the animation data from the second object to the first. Note that you must change the current frame of the animation for the cache to update, change to take effect!
This method works for rigs as long as the names of the bones are the same! Please note that the same animation data may not affect two different rigs the same way, if they are not identical in bone rotation, location, and scaling.
(Also, please note that this may not work for complex objects or movements, like armatures or movement that effects individual vertices, unless they are identical. Blender will do it's best, but on objects or armatures with different geometries, it does not know which vertices to effect unless the movement is simple directional and rotational movement applied to the object as a whole. Your best method would be to map the data with a modifier.)