  We study the generalized symmetry structure of three known discrete
nonautonomous equations. One of them is the semidiscrete dressing chain of
Shabat. Two others are completely discrete equations defined on the square
lattice. The first one is a discrete analogue of the dressing chain introduced
by Levi and Yamilov. The second one is a nonautonomous generalization of the
potential discrete KdV equation or, in other words, the H1 equation of the
well-known Adler-Bobenko-Suris list. We demonstrate that these equations have
generalized symmetries in both directions if and only if their coefficients,
depending on the discrete variables, are periodic. The order of the simplest
generalized symmetry in at least one direction depends on the period and may be
arbitrarily high. We substantiate this picture by some theorems in the case of
small periods. In case of an arbitrarily large period, we show that it is
possible to construct two hierarchies of generalized symmetries and
conservation laws. The same picture should take place in case of any
nonautonomous equation of the Adler-Bobenko-Suris list.
