"Actuaciones" are usually either the "records of proceedings" or the "proceedings" themselves, i.e. any steps, motions, pleadings, decisions, orders, actions, etc. made by the parties or the judicial authorities that are entered in the court records.
In this case it appears to be related to a motion filed by an appellant.

Perhaps you might want to try something like this:
"Serve notice on the appellant firm asking it to submit the following information in order to move forward with these proceedings /or/ in order to act on the motions/pleadings herein"

dar curso > to process, act on, proceed, handle, deal with, act in accordance with, decide...