IMAP email downloaded to Thunderbird first do not show up on iPhone / iPad, and visa versa.

I recently changed computers (Windows 7 to Windows 10) and installed IMAP accounts on Thunderbird. I also use an iPhone (6) and an iPad (A1416). If I access new email from Thunderbird first, these messages show up for an instant on either Apple mobile device, and then delete. And almost the same is true the other way - anything accessed on the iPhone / iPad first does not show up at all in Thunderbird, but is available to both Apple mobile devices. I have search the Mozilla help, on-line forums and other sources to no avail. I have also rechecked all email account settings and everything seems ot match up. This is NOT about synching calendars and contacts. Any ideas would be appreciated.