Facebook Messenger now offers Dropbox integration, allowing you to send photos, videos, and other files without ever leaving the messaging service. You can access Dropbox from Messenger's More button (the rightmost button in the conversation view). From there, select Dropbox as the source, and if you're already signed into the cloud sharing service, you'll be able to see your files directly within Messenger.
If you're sending a video or image, you will get an in-line preview in Messenger, but you'll have to switch back to the Dropbox app for any other type of file. The integration is now live on Android and iOS.