For stars around 13th magnitude, are their entries in various catalogues usually linked, or does one do this manually?
If you search by position in the CDS Simbad database by position it will give you entries in all three of these catalogues. If you search by name, then it will look for matched objects in these three catalogues. I am unsure what the matching algorithm is - it must cross-correlation on the position and look for objects within an appropriate matching radius, taking account of the proper motion of the object and epochs of the various catalogues.

This will not be foolproof, but the density of 13th magnitude objects on the sky is low enough that false matches are highly unlikely.