Regarding gmail account

Go to Settings > and click on Forwarding and POP option. 

You can provide the email address to which the mail will be forwarded.

But I want to transfer exsiting mail which are over loaded in Gmail. In your opinion new mail will be forwaded to provided E-mail Address.

Create Filter 

type your email address in To field

in next step give the id to which you want to forward the mail and select "apply to all conversation that matches the search"