Professor I applied to is about to move – can I ask them to consider my application for the new institution? I am in the process of applying for graduate programs at various universities. Given the importance of research in some graduate programs, I have been searching for professors/potential mentors when I came across a particular professor whose work impressed me immensely. I got in touch with the professor to discuss my application.

Now this is where the tricky part begins. The professor informed me that they will be moving to another university shortly and will not be accepting mentees anymore at their old university. Luckily, I had submitted my application for their new university a few days back. Is it okay to ask them to consider my application as a part of an application different from the one I had initially approached them for? My desire to carry work under them is genuine but will doing something like this make me look too opportunistic of unscrupulous?
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.