I have a nee similar to yours. I looked at the documentation for WinSCP and fund this list of commands:
The command to delete a remote file is rm. Then if you scroll down further you'll see an example that transfers a file from the FTP server to the local machine (then transfers it back). You can use this example, minus the second part. Just use the rm statement after the get command and before the close command.
Also, since you want o transfer many files, use the mget command with *.* instead of "get" and use the rm also with the *.* wildcards.
I wish that this could work for my situation. In my case files are continuously added to the FTP server, so if I use rm *.* I might delete files that came in after the mget process started and were not transferred to the local machine. Good luck, Salvador