I'm behind on messages and emails.

What does 'behind' mean? Is it okay to replace 'on' with 'with'?

That means that there are some messages and emails that you should have written, read or replied to by now but haven't.

I'd stick with 'behind on', but 'behind with' is also used.
behind: in arrears; late or slow in the discharge of obligations; "behind with the rent"

behind in, behind on, and behind with are all used.

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'Behind' here means you have a lot of messages and emails to reply to and it may take you some time to catch up.

In the same way you can get 'behind' in your work, meaning the number of things you have to do is building up faster than you are completing them.

It is better to say 'behind on' here, but if you add 'my' then 'with' is more natural. 'I am behind with my messages and emails.'
 CalifJim's reply was promoted to an answer.
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