From experience and testing, I know that the fill & dump method can be helpful if used in conjunction with the traditional constant flow method but, if dumps are too frequent, it wastes more water. As has been pointed out, fixer rinses out by diffusion. If you dump water too soon, your are dumping water that could have taken a lot more fixer. That said, I do totally drain my print washer a couple times during the wash but I have well water and have no shortage.

Many years ago, before I had a print washer, I experimented with print washing. I found a method where I stacked trays in a waterfall configuration where prints were first placed in the bottom tray and then transferred upstream when more prints were added. What this accomplished was it reused water from the upstream tray which had cleaner prints in it. The top (last) tray was fresh water. This was a rather cumbersome setup but it worked extremely well and test proved it. In fact, this principle is (was) used in a least one commercially available print washer in which the water moved sequentially from one chamber to the next, reusing the water for prints that had more fixer on them. It's a good idea but, in practice, it could involve more print handling. For many years, I use regular print washers which are easier to use in that you load them once and that's it. There is no moving of prints around.

Two ways to greatly reduce fiber-based print washing times is to be sure to use hypo clearing agent and to give the print a good rinse before it goes into a washer. I spray them off with a hand-held sprayer before putting them in the washer.

As other's have pointed out, the Ilford method is another option. I have used it but I still prefer the traditional 2-fixer bath procedure.

For film, I use one of those fill & dump washers which wastes water but I don't have a shortage of water.

I agree with those here who say that the most efficient method of washing prints is done in a print washer with a low flow rate and a periodic dump and refill, especially near the beginning of the wash cycle. If I lived in an area where water supply was an issue and I didn't have a print washer, I would use the Ilford method and hypo clearing and then rinse the print thoroughly followed by a series of water baths where the prints were allowed to sit between dumps. Agitation is fine but there is no need for constant or violent agitation. Just make sure the prints are not sticking together. I would dump more frequently during the first soaks. Since I have a print washer, I would essentially do the same thing except that I would use the compartments of my washer instead of trays.

For film, I would do essentially the same thing. In fact, that is basically the old Kodak recommended method for washing film and prints when water is scarce.