In Gmail you can choose to either forward all incoming emails or only forward the ones that match a set of rules you define.
To set up automatic forward of emails in Gmail:
Go to the Gmail account you want to forward messages from.
In the top right, click Settings (the gear icon)
Click the Forwarding and POP/IMAP tab.
In the Forwarding section, click Add a forwarding address.
Enter the email address you want to forward messages to.
Click Next and then Proceed and then OK.
A verification message will be sent to that address. Click the verification link in that message. If you are using Parseur, the verification message will appear in your document queue of your parser.
Then, if you want to forward all new incoming emails:
Go back to the settings page for the Gmail account you want to forward messages from, and refresh your browser.
Select Forward a copy of incoming mail to.
Choose what you want to happen with the Gmail copy of your emails.
At the bottom of the page, click Save Changes.
Or, if you want to forward specific new emails:
In the search box in Gmail at the top, click the Down arrow.
Enter your search criteria that will match the emails you want to forward. If you want to check that your search worked correctly, see what emails show up by clicking Search.
At the bottom of the search window, click Create filter with this search.
Tick the Forward it to box and select the forwarding email address from the list
Click Create filter.
Once enabled, the automatic forward rule described above will only apply to new incoming emails. That means you cannot forward emails already sitting in your mailbox. That can be a real problem if, for instance, you're in the middle for a marketing lead campaign and have already received hundreds of emails.
What is the daily sending limit for gmail accounts?
Gmail enforces a daily sending quota and may prevent you from sending all your emails at once:
For personal Gmail accounts, Google will limit you to sending 500 emails / day (reference).
For professional Gmail G Suite accounts, the limits are more generous. Google will limit you to sending 2000 emails / day (500 for trial accounts) and 10 000 auto forwards / day (reference). Using the Multi forward extension will count towards decreasing your sending limit, not your auto forward limit.
Limits stated above can be further restricted if you email account is brand new, as gmail wants to further prevent spam from newly created accounts.