Gmail's built-in email forwarder imposes several limitations. It only lets you forward messages to an email address that you have verified earlier. You cannot create a filter inside Gmail and auto-forward emails to addresses that are not owned by you.

You cannot forward old email threads in Gmail and it doesn't support bulk forwarding. You cannot select multiple messages inside Gmail and forward them all to another address in a single step. You'll have to forward them manually, one by one.

Email Forwarder for Gmail is a Google Sheets add-on that gets around all these limitation. Here you can create a forwarding rule, just like Gmail filters, except that you are not required to verify the forwarding address. You can forward to any valid email address. 

The Gmail add-on supports bulk email forwarding as well. You can apply a common Gmail label to all email messages that you wish to forward to another address and then create a rule, with the add-on, that will forward all these labeled message to any other email address.

And unlike Gmail that can only forward new and incoming email, the Gmail Forwarder add-on can auto-forward both past and future email messages. 

The premium edition of the Gmail Forwarder lets you create unlimited rules for auto-forwarding email messages, it runs faster (every 15 minutes) and you have a higher mail sending limit.  

Email Forwarder (Free Edition) - 1 Forwarding Rule, 20 emails per day, Runs once per hour
Email Forwarder (Premium) for Gmail Accounts - Unlimited Rules, 100 emails per day, Runs every 15 minutes
Email Forwarder (Premium) for G Suite - Unlimited Rules, 1500 emails (Google Suite for Work), Runs every 15 minutes

Google will automatically reset your daily quota around 1 PM PST.

If you are experiencing any problem or issues, go to the Add-on -> Email Forwarder menu and choose Restart. If that doesn't solve the issue, please send the Debug Logs from the same menu.

For support, email amit@labnol.org or tweet @labnol

Update: The addon's functionality is now available on the web at https://emailstudio.pro - please update!