You've been using After Effects haven't you? I came to Motion fairly recently from an AE background, and ran into the same problem. I don't recall a solution being found, aside from
1) copying the properties in the inspector (scale, position etc) from each keyframe and replicating those setting on the object you want to copy the animation too. I recall lots of post-it notes with little numbers written on them.
The best advice I can give you is to
2) keyframe the layer, rather than the object/asset, so that everything within that layer has the same properties (keyframes included)-(this can cause problems when the layer size doesn't match the asset size you are working on). You can then duplicate the layer and change the content inside it if you wanted, say, the movement but not the opacity keyframes from one layer on your new asset.
I hope that makes sense, I should leave tutorials to the experts, and I am renowned for doing everything the hard way, so if anyone can correct me, please do!