Its very easy to do:
Step1: Email account from which you have to fetch of all emails
a) Go to top right of page. Click on ‘Settings‘.
b) Under ‘Settings‘, click on tab names ‘Forwarding and POP/IMAP‘.
c) Inside ‘POP Download‘ click on the ‘Enable POP for all mail (even mail that’s already been downloaded)‘
d) Click on ‘Save changes‘ button at bottom.
Step2: New Emails account in which you have to import all emails from old account:
a) Go to top right of page. Click on ‘Settings’.
b) Under ‘Settings‘, click on tab names ‘Accounts and Import‘.
c) Now click on very first button named ‘Import mail and contacts‘. It will open a pop-up window asking your username first then your password. Its very simple just keep on filling information and clicking continue.(NOTE: Allow pop-up here if your browser is blocking it)
d) You are done. All emails backup will be done in back-end. Time for backup depend upon the number of emails you import. In my case there were around 5550 emails and it took 2 days to import all emails.