only there is one subtlety, ate the box on mail.ru:
after installation and connection box, go to account settings, server settings, untick "leave a copy on the server — not more than 14 days", or simply put "leave a copy on the server", or "not yet removed them." It seems to me that the checkbox "leave a copy of messages on server" must always be, and about the variations up to you.
By default, thunderbird download messages from server deletes them after 14 days, which I think is not very good.
So I had at least on Windows and mail.ru for gmail and yandex everything is fine.
How will MAC OS do not know, but worth checking out.