Of course there is no such thing as reincarnation, because there is no such thing as a "spirit" or "soul". That was and is a primitive invention to explain the existence of a "mind" that seemed to be "inside" a body—whereas, of course, "mind" is actually a function OF the body.
The practitioners of Zen actually have discovered the truth, but without realizing it! They turn off the surprisingly minor "awareness" function as much as possible and operate as an automatic body.