As discussed in other answers, the easiest option would be to just have a custom rope and then everything works as you would expect. In a similar pro-climber vein, don't discount using rigging from the camera crew or stashing gear / fixed ropes. Note that these options incur a penalty in terms of either cost, weight of ropes to carry to the crag, or the added weight of a tag line (which can be quite significant on hard routes).
More common (sometimes even required at sport crags for mere mortals) is to lower off using intermediate anchors. In the video of Ethan Pringle on Jumbo Love, you can clearly pick out intermediate anchors (e.g. at the 15:27 mark next to his right foot), so option 1 or 2 below would likely be the method of choice. Note that we're fully into the land of non-beginner rope work and shenanigans.
Engage brain and receive training before doing anything that follows
With all of the options to be discussed, the climber should take steps to make sure that they do not drop the rope! Just as with cleaning a sport anchor, there are many ways to achieve this.
A workflow would be something like....
Option 1
Belayer lowers climber to next highest anchor. This may require the climber to "tram" along the rope on steeply overhanging routes.
Climber securely connects to anchor. On a project route, you would likely have a couple carabiners left at the intermediate anchor for just this purpose.
Belayer disconnects from the rope. !Make sure the climber is safe first!
Climber pulls the rope, either by:
1) Pulling from the climber end, taking the belayer end all the way through the top anchor, which then hopefully falls to the ground for the belayer.
2) The climber unties, pulls the the belayer side of the rope such that the climber end goes up to the top anchor and back down to the climber at the intermediate anchor. This method is generally easier.
Regardless of how the rope was pulled, the climber threads the intermediate anchor, the belayer puts them back on, and the belayer lowers the climber.
Repeat if necessary.
Option 2
Belayer lowers climber to intermediate anchor.
Climber secures to intermediate anchor.
Belayer takes the climber off; climber pulls the rope through and threads the intermediate anchor.
Belayer takes the free end of the rope, which has come down to the ground, and ties it to a second rope.
Belayer lowers the climber from the intermediate anchor, perhaps with shenanigans to pass the knot; or the belayer clips a belay device to the rope and the climber tags up the second rope & belay device and rappels from the intermediate anchor with two ropes.
Option Danger
For a rather more daring/dangerous/please-don't-do-this approach...DANGER:
Lower the climber to a quickdraw + bolt.
Use the quickdraw to connect the climber directly to the bolt (having one extra quickdraw makes this much easier).
Belayer gives a touch of slack, climber connects to the strand going directly to the belayer with a locking carabiner, e.g., by clipping to a figure-eight-on-a-bight.
Climber unties the rope from their harness, pulls this newly freed end through the quickdraws above and back down to them. If the bolt that the climber is hanging on were to fail, they would take a leader fall onto the locking biner. Not the safest thing, but at least there's some level of backup...
Climber ties back in, belayer takes up the slack, climber unclips from the quickdraw, belayer resumes lowering from the quickdraw that the climber was previously connected to.
Some people have even been known to just clip into the quickdraw and not bother with the extra hijinx of connecting the the belayer side with a locker. Regardless of whether this step is done or not, there will be a least a couple moments where the climber is hanging off a single bolt and a single carabiner. This breaks one of the cardinal rules (never trust your life to a single piece of gear), but new bolts in sound rock and quickdraws are usually pretty strong and...
If you can't see why this is extremely dangerous for yourself, don't ever try it.
Option Free Solo Belayer
While not directly applicable to the routes mentioned, the following can often be quite useful. This would generally apply to a route where just an extra 5-10m of rope are necessary to get the climber down from hanging in free space and the route starts with easy terrain (say 10m of 4th/easy 5th before the route steepens).
Belayer lowers climber as far as possible, to the point of having a stopper knot on the brake side of the belay device taking up the load (or even better, having the belayer tie into their end of the rope before the climber ever leaves the ground).
Belayer climbs upwards on easy terrain, acting as a counter-weight to the climber. Once the climber reaches the ground, they untie.
The belayer downclimbs back to the ground, being cognizant of the fact that they are effectively free-soloing. Herein lies the importance of the route starting with easy terrain.
Occasionally, on overhanging routes where the ground slopes away from the base of the route, the climber doesn't need to untie, but rather just walk uphill back to the base of the route as the belayer downclimbs / gets lowered as a counter-weight.
Research & Development Option
One product scheduled to hit the market in 2018 is the Beal Escaper, additional video and commentary. The device connects to the anchor and then one end of your rope is tied to it. After doing a full length, single strand rappel, simply do ten hard tugs and everything comes falling down to you. Shades of Samwise Gamgee and his elven rope.
Seems simultaneously fascinating and terrifying. The main utility would seem to be multipitch routes, where an entire second rope (an extra 5-10lbs) has to be carried up the route just to use as a tag line for rappelling off. In the context of the original question, this would still require the climber to a) get lowered and secure into an intermediate anchor; b) pull the rope through after being taken off bely; c) either tag up or carry along a belay device and the Beal Escaper.
As this isn't scheduled to hit the market until the end of 2018, it hasn't undergone "real world" testing. On a first pass, it seems absolutely terrifying and prone to many "what if?" scenarios. On the other hand, Beal is a well respected brand and they would have to have tested it extensively before even thinking of selling such a product commercially...
Canyoneering-World Options
Moving well outside the realm of the question, you can look to the canyoneering world for inspiration on how to do long rappels with a minimal amount of gear. For that community, they aspire to "ghosting": traveling down a canyon leaving absolutely no gear behind (not even a sling around a tree or, heaven forbid, a bolted anchor). They make the sketchy anchors used while alpine climbing seem OSHA approved in comparison.
For examples, there are things like the Imlay Canyon Gear Fiddlestick (video) for retrieving a rope with only a tiny pull cord after a full length rappel, or the SandTrap which is effectively a tarp that you fill with sand, rappel off of, and then use a pull cord on one corner to dump all the sand out of and send everything down to you.
Again, great ways to make rappelling even more dangerous and not applicable to the tame world of sport climbing.
