Cell Phone Email app?

I don't use t-bird, however look through the email settings on your computer and see if you can locate "store on server" or similar as many emails have this option.
Oftentimes computers have more options than phones...

THe problem is in your thinking.. When you download the E_mail to your home computer and remove it from the "HOST" ushing protocol the file now exists on yoru comptuer.. NOT the host

Your phone can not access it

SOLUTION
Dump the Client based E-mail (T-Bird) and go with Web-Mail. Yahoo, G-Mail. Microsoft's current mail system and most other E-mail hosts today offer a Web mail.

With Web mail I can access from this Chromebook, the one beside it, the Windows machine below it, the one in the ham shack, the one in the bedroom My Phone or tablet. All see exactly the same thing.

If I move it off to a folder.. all see that folder

All see exactly the same thing

NOTE: the downside: Limited storage on the free Web clients. pay for more is an option.

The app "K9 Mail" Just like Thunderbird on your PC. Free app.

I use Tbird and the Blue Mail app. I just checked and things I saved on Tbird are on my phone. I do go through Gmail and download to Tbird.

Depends on what phone app you're using. Many times they don't autodiscover all the folders available on the imap. They just setup common ones like inbox, sent, deleted.

There should be an option in the app to download additional folders from the server, or another way to describe it is which folders to sync with the server.