IME when fish run they go downstream much more often than up. Who wants to pull a 15 to 25lb fish back upstream? Also it keeps the line from angler to fish short or at least as short as possible. That reduces current pressure on the line and clears more water for other anglers to fish.
If the banks are clear of anglers and you want to stay in one spot go for it.
Nope. Salmon almost always run upstream when hooked in a river, unless they're hooked in the tail. Most fish tend to run in the opposite direction of the pressure you're applying.
The key is to keep side pressure on the fish rather than holding your rod tip high. With the current carrying your line downstream, the side pressure encourages the fish to run upstream toward you.
We see so many people fishing on the Vedder, but unfortunately, less of them know how to fight a fish. That's why you see so many anglers walking downstream fighting fish—some quietly, others so noisy for the entire river to hear.
There is one important exception. When fishing the canal, some dude float drift too far downstream before hooking a fish, there is a tremendous amount of line to recover. At that distance, the line angle can make side pressure ineffective.
In that situation, walking downstream actually makes sense. It helps recover line, saves everyone else's time, and reduces the chance of tangling with other anglers' floats flying into the river like nobody's business. But hey, don't drift too far when dozens are downstream