  We extend the notions of multipole and subsystem symmetries to more general
{\it spatially modulated} symmetries. We uncover two instances with exponential
and (quasi)-periodic modulations, and provide simple microscopic models in one,
two and three dimensions. Seeking to understand their effect in the long-time
dynamics, we numerically study a stochastic cellular automaton evolution that
obeys such symmetries. We prove that in one dimension, the periodically
modulated symmetries lead to a diffusive scaling of correlations modulated by a
finite microscopic momentum. In higher dimensions, these symmetries take the
form of lines and surfaces of conserved momenta. These give rise to exotic
forms of sub-diffusive behavior with a rich spatial structure influenced by
lattice-scale features. Exponential modulation, on the other hand, can lead to
correlations that are infinitely long-lived at the boundary, while decaying
exponentially in the bulk.
