I'm only adding another answer here, because I immediately thought of a two philosophers that no one else has mentioned.
"A person is not a Thing, not a substance, not an object"
A quote from your question:
"We all know that processes are not solid and change all the time, yet in this particular process there is a nagging sense that there is a solid permanent “me” hiding in that process somewhere."
For Heidegger, this is not so. Some of his philosophy could be summarized as saying - There is Being and Being is unconcealed in a particular fashion in accordance with certain phenomenological constructs. For example, 'Being and Time', perhaps his magnum opus, attempts to understand Being in terms of the relationship between Being and Time. Throughout his work, Being can never be found as having some SELF that exists independently of the relationship between Being and the clearing that provides a context for Being to occur as it does. In some ways this is similar to Dependent Origination (Pratītyasamutpāda) in Buddhism.
In addition, per Wikipedia:
Heidegger's "Dialogue on Language", has a Japanese friend (Tezuka Tomio) state that "to us [Japanese] emptiness is the loftiest name for what you mean to say with the word ‘Being’”
From the same Wikipedia article as the Heidegger quote
Nietzsche (as well as Buddha) accepted that all is change and becoming, and both sought to create an ethics which was not based on a God or an Absolutist Being.
and further ...
The similarity between Nietzsche's view of the Ego as flux and the Buddhist concept of anatta is also noted
While Buddha proposed the rejection of cravings - the triumph over desire as it were - as a path to the cessation of suffering - a path to cutting off 'the endless cycle of birth and death (Samsara)', Nietzsche offered up the myth of Eternal Return in The Gay Science, saying:
What, if some day or night a demon were to steal after you into your loneliest loneliness and say to you: 'This life as you now live it and have lived it, you will have to live once more and innumerable times more' ... Would you not throw yourself down and gnash your teeth and curse the demon who spoke thus? Or have you once experienced a tremendous moment when you would have answered him: 'You are a god and never have I heard anything more divine.'
In what is perhaps an inversion of the Buddhist doctrine of rejection, Nietzsche proposes:
My formula for human greatness is amor fati: that one wants to have nothing different, not forward, not backward, not in all eternity. Not merely to bear the necessary, still less to conceal it—all idealism is mendaciousness before the necessary—but to love it.
Though on a surface level, the parallels are subtle, after investigation, it becomes clear that the idea of no-self is embedded in both Samsara and Eternal Return equally. In both instances, there is an experience subject to past actions ( when are past actions created? ), and that experience repeats eternally. There is no self experiencing Samsara that is separate from Samsara. The illusion of Samsara is perpetuated by desire as an endless cycle of grasping and rejecting; of becoming and termination. In as much as one is at odds with the present moment - either by craving or suffering the consequences of previous actions - one experiences the self as an entity separate from the flow.
The cessation of desire is achieved in Buddhism by cutting off desire through negation ( though negation is but one method in Buddhism ), and with Nietzsche, by amor fati. With amor fati one is not afflicted by desire as one loves their fate and embraces the circumstances of a given situation regardless of how bad it is; as such, they wish to change nothing. In that space, one simply is - and as such, that is-ness is not separate from the flow. Nirvana is Samsara.