Currently, the "Move Android APKs to a different release track" build step expects that the user inputs the version codes, or provides the previous APKs.

Inputting the version codes, whether by parameter or whatever, is a bit awkward. Similarly, providing the previous APKs requires that they were previously archived somewhere (which hopefully people do anyway), and then using Copy Artifact to get them into the current workspace.

So there should be an additional "Use APKs uploaded in another build" option.

This would provide a field for the job name, plus a dropdown with stuff like "last successful build", or "upstream build" etc.

Hopefully these dropdown values and UI are somewhat standardised, or provided via extension point?
See the Copy Artifact plugin, as it probably has the most useful set of options in this dropdown.