How do I back up a lot of emails onto a flash drive?

I am helping my family member save their emails to a flash drive or hard drive, how do I do this properly? And contents of those emails. A lot of them are from loved ones that aren't around anymore. I'm doing this from 3 their email accounts, how do I keep track of what I have saved and what I haven't saved, there are thousands of emails. I hope this is the right place to ask, thank you guys!