Unless I'm out of my gourd, World of Warcraft has a feature like this. If, at the end of a dungeon, the party leader goes into their version of the Duty Finder and queues for pretty much any other dungeon (within their level range--their Duty Finder doesn't let them queue for non-current content stuff, you gotta walk to the entrance for anything else or get thrown in there via roulette), it will prompt everyone else in their group to accept or decline the queue. If someone declines, cool, wait till they leave or get kicked and queue again, and your party will automatically get filled.

Might be a holdover from WoW's ancient partying system, but something like it wouldn't be out of place in FFXIV.