  We develop a general theory for Markov chains whose transition probabilities
are the coefficients of descent operators on combinatorial Hopf algebras. These
model the breaking-then-recombining of combinational objects. Examples include
the various card-shuffles of Diaconis, Fill and Pitman, Fulman's
restriction-then-induction chains on the representations of the symmetric
group, and a plethora of new chains on trees, partitions and permutations. The
eigenvalues of these chains can be calculated in a uniform manner using Hopf
algebra structure theory, and there is a simple expression for their stationary
distributions. For an important subclass of chains analogous to the
top-to-random shuffle, we derive a full right eigenbasis, from which follow
exact expressions for expectations of certain statistics of interest. This
greatly generalises the coproduct-then-product chains previously studied in
joint work with Persi Diaconis and Arun Ram.
