  The $n$th term of an automatic sequence is the output of a deterministic
finite automaton fed with the representation of $n$ in a suitable numeration
system. In this paper, instead of considering automatic sequences built on a
numeration system with a regular numeration language, we consider these built
on languages associated with trees having periodic labeled signatures and, in
particular, rational base numeration systems. We obtain two main
characterizations of these sequences. The first one is concerned with $r$-block
substitutions where $r$ morphisms are applied periodically. In particular, we
provide examples of such sequences that are not morphic. The second
characterization involves the factors, or subtrees of finite height, of the
tree associated with the numeration system and decorated by the terms of the
sequence.
