If you want to mass forward existing emails already sitting in your Gmail mailbox, here are some options:
Option 1: install the Multi forward for Gmail Chrome extension (free)
The Multi Forward Chrome Extension will add a new action button that will allow you to select emails in your mailbox, enter a forwarding email address and individually forward each email to the given forwarding address. Head over to the Chrome Store to install the Multi Forward for Gmail extension and have a look at the supporting documentation.
Note that 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.
Option 2: Use Multi Email Forward solution (commercial)
If the extension above is not sufficient for the volume of emails you have to forward, use the Multi Email Forward solution by cloudHQ. Multi Email Forward will go around Gmail sending limits by working in the background forwarding the emails. This solution is not free, but it is one of the best choice if you have a large volume of emails to transfer. More info: https://www.multi-email-forward.com/