What can I do to seal up these openings between our cabinets?
That isn't "unfinished work". Cabinet makers almost never fill that gap because 1) its size and shape may change when the cabinets are installed, and 2) no one ever sees it (unless they have kittens).

In your shoes I'd get some heavy card stock or rolled sheet metal and cut it to fit. Use carpet tacks, a stapler, or maybe hot glue to mount it to the underside of the cabinet frames. Keep the front edges back 1/4" from the front face of the cabinet to keep it hidden. 

I'd guess that kittens won't try as hard as mice would to get in there, so card stock should work and will be easier to handle.