  For non-uniform cellular automata (NUCA) with finite memory over an arbitrary
universe with multiple local transition rules, we show that pointwise
nilpotency, pointwise periodicity, and pointwise eventual periodicity
properties are respectively equivalent to nilpotency, periodicity, and eventual
periodicity. Moreover, we prove that every linear NUCA which satisfies
pointwise a polynomial equation (which may depend on the configuration) must be
an eventually periodic linear NUCA. Generalizing results for higher dimensional
group and linear CA, we also establish the decidability results of the above
dynamical properties as well as the injectivity for arbitrary NUCA with finite
memory which are local perturbations of higher dimensional linear and group CA.
Some generalizations to the case of sparse global perturbations of higher
dimensional linear and group CA are also obtained.
