Below are some pointers about fingering in shakuhachi playing. They are based on my own journey and are not prescriptive.
The most important fingers are the middle fingers and the right-hand thumb! They are essential in balancing the shakuhachi when moving between different fingerings. They work in conjunction with the chin to create a complex interplay of weights and fulcrums. Try playing an easy piece and feel how they interact when the fingers over the holes move.
Keep all fingers at a reasonable and even height above and below the shakuhachi. Flautists and other instrumentalists develop impeccable finger technique by economising on finger movement. The rigour of this type of practice may not be fully applicable to shakuhachi playing. However, there is something to be learnt from occasionally watching what our fingers do.
Observe how the wrist reacts to partial covering of holes. The fingers should make most of the movement. If the wrists move too much, it can cause other fingers to move and partially open a hole. This is especially true when learning to play on longer shakuhachi. The location of this problem finger is often where we least expect and further up the shakuhachi. Don't forget that partially-covered holes are accompanied by the meri-kari technique. Inconsistent lowering or raising of the head can affect the hand and finger positions.
Practise the three types of partial covering of holes: hankai or half opening, han-on and kazashi. Play an exercise moving back and forth between adjacent notes like re to chi-meri (hankai), ro to tsu-meri (han-on) and chi to ri chi-meri (kazashi).
When moving some fingers up and others down, concentrate on the finger lifting off the hole. This finger starts the chain reaction while the finger going down relaxes. If you hear 'blips' in your fingering that you had not intended (sometimes they sound good so remember how you did them!), it is usually because the finger going down arrived before the other finger had time to move.