Facebook has spent a lot of time beefing up app integration inside its Messenger platform, adding support for services ranging from Uber to payments to goofy animated images to lighten the mood.
With both apps updated, the next time you're in a conversation and want to send a file, tap on the "More" button (three-dot icon just above the keyboard). Find and select Dropbox from the list of services, and tap on Open. The first time you select Dropbox, you'll have to grant permission for Facebook Messenger to access your files. Once that's done, select a file you want to send and let Dropbox do its magic. A few seconds later, Messenger will open with the selected file attached to your conversation.
The recipient receiving the file can view the file in Messenger, or open it in his or her Dropbox account.
Facebook states that the feature is rolling out, so if even after updating you don't see Dropbox listed as a service, keep checking the list.