All files saved using Mapsource can be opened using Basecamp, Although I have to say they could have made it easier to do. If you have both Mapsource and Basecamp installed on your computer, there are a couple of methods to transfer stuff :-
1) Copy and Paste
Go into Mapsource and open your desired files, e.g. File >> Open >> [file type MPS, GDB, GPX, KLM etc] >> highlight the items you wish to transfer >> Copy >> open Basecamp >>Edit Paste
2) Open File
Using File Explorer open the folder where your Mapsource files are stored >> right click on a file >> select Open With >> Basecamp
If Open With >> Basecamp is not an option select Open With >> Choose Program and select from a browse list
EDIT (Added 15MAR2017)
3) Import into My collection (Basecamp)
While in Basecamp goto >> File >> Import Into My Collection >> navigate to the folder containing the files you want to open >> Click on file (or Control Click multiple files) in order to highlight them >> OPEN