I thought about how a continuous (in time and space, but not in states) cellular automaton could look like. The most straightforward generalization which came to my mind is the following:
Let $(X,*)$ be a group and $(X,d)$ a metric space such that all the maps $y\mapsto x*y$ are $d$-isometries (for example Euclidean space with translations), also let $S$ be a discrete state space. Now I define an infinitesimal rule on $X$ as a family of transition functions $\{\Gamma_s\}_{s>0}$ with $\Gamma_s:(B^d_s(e)\to S)\to S$ such that $$\Gamma_{s+t}(f)=\Gamma_s(x\mapsto\Gamma_t(y\mapsto f(x*y)))$$ for $s,t>0$. We could restrict the $\Gamma_s$ to something like Borel measurable sets, but I think that's not necessary since we can always introduce some kind of error state $\epsilon\in S$ so that a cell falls into $\epsilon$ if it sees a pattern it cannot handle (like Borel non-measurable).
My questions, does this make any sense, are there any non-trivial infinitesimal rules known on e.g. $\mathbf{R}^2$?