Unfortunately, that sounds like a terrible idea. What if Mongo works on two machines at the same time? Both databases are likely to be corrupted. A better way would be to write a script that used mongodump and mongorestore to dump the database into Dropbox and restore it from the dump. However, you will have to run these manually.
The reason you do not see any changes to the database files is probably due to the fact that Mongo predefines its database files, so their size never changes, but only the contents inside. Dropbox may not detect this. Mongo also does not write to disk immediately. Rather, it uses memory-mapped files that can later be reset (this should be a matter of seconds, although this should not be the reason).
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