Receiving letters on the server with yandex, mail, etc. for local clients?

Question

Mail server

Hello.
Most likely I did not understand the question.
The question is:
Is it possible to configure the receipt of letters on the server from mailboxes (mail, yandex, gmail ...), so that clients on the local network would contact the local server for received letters?
And it turns out that the working machines, each for himself receives new letters, stores them on his hard ones, after sending the letter to one, others have to pump it up separately. Yes, and 5 mailboxes.
I hope I asked clearly)

Answer 1

If I understand the question correctly, then no. If you use other people's mail servers, then mail traffic goes through them.

Answer 2

procmail is able to collect mail from different mailboxes and lay out, forward according to the rules

Answer 3

of course, IMAP + file
for example
https://www.howtoforge.com/how-to-migrate-mailboxe...