Well, if your target database is reasonably modern, it probably supports LOAD DATA INFILE syntax, which allows you a very fast way to load a CSV file (for example) into the database. You could query your database A and then use Text::CSV_XS to generate an appropraite CSV file.

Move the CSV to the target (database B) machine and run the LOAD DATA INFILE. All of that should be reasonably scriptable, and probably the fastest route.