So the answer to your question about moving is definitely a personal thing, but I'll just share my experiences and perspective…
I find that most anglers like to camp out in one spot and fish a hole. This clearly works better on larger rivers and big water. Sometimes the fish are smarter, bigger, and the hatches on big water are definitely more specific. Feeding patterns are more erratic. Early morning and evening tend to be the best for this.
BUT if you are fishing tenkara, arguably the best water for tenkara is small streams.
When fishing small streams, trout tend, generally speaking, to be way easier to spook, and yet way less selective on what they eat.
A big meal is rare and they go for big insects, bigger flies than the average fisherman would guess, and they don't hesitate to slam your fly.
So I have found that moving up stream, being careful not to cast a shadow on the water, and by walking AS LITTLE in the water as physically possible (to disturb and scare less fish) the more you will catch.
Sometimes the fish are concentrated in one pool, yet two picture perfect pools above and below that will be practically empty. This happens to me all the time, and is one of the reasons I'm a huge proponent of walking up the stream fishing as you go. If I don't get a hit or hook a fish in a pool after 3 good casts to each lie, I move on. No sense in beating a dead horse.
Plunge pools and pocket water are the best for tenkara. Stand at the base of the pool, cast your fly up and into the waterfall/plunge, and let the water sink that fly. Use a bead head if you aren't a purist. Make sure the fly is really buggy and ugly, around size 12 or 14. Something like an ausable ugly is PERFECT for wild trout.
You can fish dead drift, or jerk the fly a few times to get attention. IT WORKS.
I've fished Tenkara in Japan and they never stand in one place fishing one hole, and in fact, they rarely change flies.
I use two flies in 99% of my fishing – Asuable ugly for under the surface, ausable bomber for surface fishing. And when they are rising but nothing else works, I throw on my third fly… the usual. The rest of the time I fish Kebari that I tie myself for fun, but the success rates are slightly lower :)
Tight lines and enjoy Tenkara. Feel free to pick my brain if you'd like, I spend WAY too much time fishing tenkara instead of doing what I "should" be doing.