Professor I applied to is about to move – can I ask them to consider my application for the new institution?
It is absolutely not a problem for you to ask for help!

Faculty move is part of academic life and often times one of the trickier things to handle – especially when figuring out how to handle relocating a lab full of people. Your interest in their work is not tied to where their lab currently is, at least as far as you describe it, so it is fully acceptable for them to advise you on the application process. 

Regardless of their affiliation, their role has been strictly advisory (as described). This means it is as if one professional is simply giving advice to someone entering the field – there is no conflict of interest.

Continue, and if that professor’s research excites you, you should certainly try to get into their new university and rotate with them to see if it’s a match.