Moving files using "Rename, Move, & Copy files..." is the recommended method of moving files.
"Live time-shifting files" - yes they are in a folder with many different files. If you choose to record in JTV format, these files are saved as your recordings. If you choose to save time-shifting when you end a time-shifting session, the folder gets saved as well. In both cases, there is a .jtv file in the folder. This file is the stub file that "represents" the recording in MC. So in MC, select the .jtv files, and run "Rename, Move, & Copy..." tool. The recordings should be moved correctly.