What will Dropbox do when I boot my external backup drive?

My MacBook just died and I brought it into the store to get it fixed. Luckily I have a recent backup (from last week) that I made with SuperDuper so I can actually plug it into my wife's Mac and boot it up -- the drive is a bootable clone -- and continue my work. Since I made the backup of course there were changes that I made to files, most of which were uploaded to Dropbox. When I boot up the external HD, what will Dropbox do? Will it automatically update the files to the most recent version?