Facebook Messenger users: Is there a way to have FM send me an email with the contents of each message I receive?

I don’t think so. You could just copy and paste them yourself, I guess.

You can always read them on Messenger (which is, therefore, like an online archive).

Not that I am aware of, but the messages stay in messenger. You can search messenger with key words or scroll back your messages with a specific person if you need to locate a message.

You can download your conversation anytime you want.
An email of each submission would not be very helpful.

I have a phone app that emails a copy of every Android message to my gmail account. I disagree with @Forever_Free – I use these archived messages all the time in gmail for information, URLs, etc. I find it very useful.

The problem with keeping them in messenger is that (a) it is beyond difficult to search for old messages, and (b) I don’t always have my phone at my side.

@elbanditoroso In case you didn’t know, you can access Messenger on a website: messenger dot com.

Echoing @Hawaii_Jake, I almost always access Messenger on my desktop and it is easy to look through individual conversations.

@elbanditoroso I am glad that option works well for you. Perhaps my comment came out of the sheer volume of messages I deal with. Adding an email to the mix as opposed to going to the source system to me is one of doubling down when you are searching. Then again, I architect messaging systems and know there are more convenient searchable tools in the initial system of record.
Typically a system design would make a search or output to archive option available versus a rule for autoforwarding. Even Email systems have taken autoforward to other email addresses out of their options as this is a prime point for hackers to spam or bog down the initial system creating an email storm.
I hope you find something that works for you.