How can I duplicate an animated object and then move it to another place?

Are you using the sequencer to animate it? If you are using the level sequencer it will be using the world position offset. You won’t be able to copy and paste it to a different location because those keyframes are meant for a specific world location.

Another way that I have in mind is to use the local location. In UE5 you can use a local sequencer inside your respective blueprint by adding a sequencer actor component and animate from there. I believe you need to enable a plugin to do so.

Other than the sequencer my guess will be the material itself. If this is the case, you need to analyze and adjust the values inside the material.

Here is the documentation about the actor sequencer component. Let me know if it helps.