For NTFS drives you can search for multiple media file types using Everything Search "ext:" like this:
just separate each extension with a semicolon. Everything uses MFT structures from NTFS so it won't work on other file systems. But on the thumb drive you could use Explorer "OR" search.
At the top of the drive or root folder you want to start the search, in the Explorer window search input use OR as in
*.mkv OR *.mp3
Once you have all the files in the list it's a matter of drag and drop.
If you keep a text file of various search strings then you don't have to remember the keywords.
I assume you've looked on the popular download sites. Have you also tried SourceForge
Sometimes there's a hidden gem in the open source programs. Adding Win32 to the search usually narrows it to Windows utilities.
To do all that multiple search masking from scratch is half way to a file manager. It's already been done for free too many times to duplicate. I understand the particular feature may not be there. But fast searches can make up for a lot. Even Windows Explorer non indexed search should be pretty fast if you are confined to a thumb drive, esp. if the folder tree is not very nested.
If you want to sync particular folders to each other, I'd recommend FreeCommander. It has an intuitive Folder Sync built in and is free, as the name suggests. Each tab has it's own history. Once you sync a few folders clicking on the input line for each tab pops up a history and it's quick to select each side that way. Sync has a check box to recursively include subfolders or just sync the root folder on each side.