One way would be:- create another email addressee on your isp i.e. [email protected] then when you open your emails on the laptop forward them to your new address then download these forwarded emails on your desktop
You can read them via Webmail on your laptop, and they will still be in the mailbox ready for downloading when you return.
However you do it though, be aware that most mailboxes have a limited capacity these days, and if you leave all your mails in the mailbox, they could exceed it's capacity.
One way around this is to adopt a slightly more cumbersome approach. Download your mails into your latop when away, then when back home, forward hem to yourself using Send only, rather than the normal Send/Receive. Of course you need only forward the ones you want to keep!