Welp, it looks like i'm going use the moon rasenshurikens on cultists but still likely would hang out with friends or protect them from anything by using magical chakra powers.One would assume any jutsu or technique that specifically works with someone else's internal chakra would function just fine with other people that became someone from the Naruto setting (SailorPizza for example presumably has chakra and a chakra network due to being a genderbent Naruto). It's just that if it's only people from FiC getting transformed... there just aren't a lot of us with Naruto avatars that would have chakra.
Presumably things like psychic and many magical and reality warping effects sourced internally would interact properly, or in some form, between people who were turned into things from different settings but where the powers were close enough to each other.
We're all in the same setting of the real world in this scenario so one assumes no external stuff in the environment that would be in home settings unless introduced by people transformed into their avatars.
Personally, as Shub Niggurath, if I still find myself caring enough I'd try to spread the various powers around to even the playing field for everyone which would make more things work properly. I mean Shub Niggurath is an embodiment of reproduction, mutation, and life as commonly presented. Spreading powers around seems in character even if I were to lose my humanity.
If I find anyone with chakra from the Naruto setting it probably would be gifted to others via mutation. If the God Tree and Otsutsuki clan could introduce chakra to a world that didn't have it in it's home setting I don't see why an Eldritch reality warping God couldn't give people it after finding an example to spread to others. There's even implication that the planet the Naruto setting takes place on is not unique in having chakra artificially introduced in the past.
Maybe I can create a High Priest or Chosen One with my powers who's job it is to make sure anything mutated or created by me before I leave Earth doesn't destroy the world and make sure powers from other transformed people are proliferated properly.
So... @sneakykitten and @Sailorpizza may want watch out for some possibly mad Shub Niggurath cultists seeking to acquire blood and tissue samples to spread chakra and bloodline traits to other people. The same would apply to anyone with powers that can be spread via Shub Niggurath's powers gifted to whatever cultists I leave behind.
Thanks, @V4Guss. for coolest answers.