Nearby cellular automaton rules

In a range r cellular automaton the new color of a particular cell depends only on cells at most a distance r away. One can make an equivalent cellular automaton of larger range by having a rule in which cells at distance more than r have no effect. One can then define nearby cellular automata to be those where the differences in the rule involve only cells close to the edge of the range. With larger and larger ranges one can then construct closer approximations to continuous sequences of cellular automata.