  We develop techniques of mimicking the Frobenius action in the study of
universal homeomorphisms in mixed characteristic. As a consequence, we show a
mixed characteristic Keel's base point free theorem obtaining applications
towards the mixed characteristic Minimal Model Program, we generalise
Koll\'ar's theorem on the existence of quotients by finite equivalence
relations to mixed characteristic, and we provide a new proof of the existence
of quotients by affine group schemes.
