There is no possibility for transferring or backing up e-mails from one Gmail account to another Gmail account currently without downloading it.
But, you can create Microsoft's outlook.com e-mail account and transfer your e-mails to outlook.com e-mail inbox.
Outlook.com also provides similar features like Gmail.
In additon you can import all of your Gmail account e-mails to outlook.com email account.
Create an outlook.com account then,
Go to settings (gear icon) in outlook.com, click on 'Import email accounts', select 'Google', then login and authorize the access.
Then, e-mails import process will happen at server side, no need to download and no band width issue.
If you like you can continue to use outlook.com as your primary mail account, even you can receive / send e-mails on behalf of your Gmail ID.
There is no possibility for
There is no possibility for transferring or backing up e-mails from one Gmail account to another Gmail account currently without downloading it.
But, you can create Microsoft's outlook.com e-mail account and transfer your e-mails to outlook.com e-mail inbox.
Outlook.com also provides similar features like Gmail.
In additon you can import all of your Gmail account e-mails to outlook.com email account.
Create an outlook.com account then,
Go to settings (gear icon) in outlook.com, click on 'Import email accounts', select 'Google', then login and authorize the access.
Then, e-mails import process will happen at server side, no need to download and no band width issue.
If you like you can continue to use outlook.com as your primary mail account, even you can receive / send e-mails on behalf of your Gmail ID.