  We consider the dynamics arising from the iteration of an arbitrary sequence
of polynomials with uniformly bounded degrees and coefficients and show that,
as parameters vary within a single hyperbolic component in parameter space,
certain properties of the corresponding Julia sets are preserved. In
particular, we show that if the sequence is hyperbolic and all the Julia sets
are connected, then the whole basin at infinity moves holomorphically. This
extends also to the landing points of external rays and the resultant
holomorphic motion of the Julia sets coincides with that obtained earlier using
grand orbits. In addition, if a finite set of external rays separate the Julia
set for a particular parameter value, then the rays with the same external
angles separate the Julia set for every parameter in the same hyperbolic
component.
