How do you mute with your right hand, especially when moving to a higher pitched string?

I'm trying to learn the "ring finger on A" muting strategy for 4 string. Thumb rests on the p pickup face, to play E string I just play, to play A string the thumb drops down a hair into the E to mute.
When playing D, the fingers that just plucked D come to rest on the A.
For the G, I am trying to get it to become automatic to place my ring finger down on A and let the plucking fingers mute D and thumb mute E
It seems like it works...

But I've been thinking this feels awkward even after a lot of practice. Like I'm getting more automatic at muting A string with the ring finger but it doesn't feel very natural. Do I just need more isolated practice on this, maybe try harder to let the ring finger float over A at all times, or what,

Is the Wandering Thumb, where you mute the next string with your thumb whichever string that is, "better"? It seems slightly more natural and would extend to 5 string easier.

It just seems strange to teach my hands a separate movement to put the ring squarely on the A without looking, for just one string, what seems to me like the least used string. Wandering thumb is more like... Just move to the string and play... Thumb follows the fingers.

When I first started learning bass I didn't focus as much on muting but now I'm looking to start slow and play clearly and cleanly. So I figured I'd ask you guys about right hand muting technique, if this is something you guys pay attention to, and for ideas, tips and tricks please.

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