Gmail offers a lot of space, handy search capabilities and universal access. You can bring all this functionality to your Mozilla Thunderbird email by importing it into your Gmail account.
Gmail offers IMAP access, a protocol that stores your emails on a server but lets you view and work with them as if they were stored locally (that is, on your device). It also turns importing emails into a simple drag-and-drop action. To copy your messages from Mozilla Thunderbird to Gmail:
Set up Gmail as an IMAP account in Mozilla Thunderbird.
Open the folder containing the emails you want to import.