Open Thunderbird. Go to Settings > Account Settings > Synchronization & Storage. Make sure "Keep messages for this account on the server" is selected and "Synchronize all messages" is enabled.

Go to Settings > Notifications. Ensure that the "New Mail Notifications" are enabled, and you can choose a sound here as well.

Thanks for the prompt answer. I was unable to fix the problem. There is no "Synchronization & Storage" in the account setting. There is a "Fetching mails" folder with no box to "Keep messages for this account on the server" to check. I downloaded the configuration from the Linux desktop application a couple of days ago. Sandro

K9 Mail, which the Android Thunderbird 8.x and 9.x client is based off of, had the concept of "folder classes". 1st, 2nd, and 3rd class, which had different configurable mail checking schedules.

This ui was removed in the rebrand and merging of K9. Nothing has replaced it. Thus, on a new install, nothing but inbox is regularly checked. No developers using the user+folder@example.com addressing scheme, I suspect.

Switching to the K9 branded client is no improvement, unless one is willing to go back to 7.x, perhaps further.

Wandering answer-givers don't even notice they are in the Thunderbird of Android forum, and thus give desktop answers.