Kinetics of phase transformation on a Bethe lattice governed by single-spin-flip Glauber and spin-exchange Kawasaki dynamics is examined. For a general Glauber dynamics for which all processes (splitting and coagulation, growth and decay of clusters, as well as creation and annihilation of single-spin clusters) take place, the addition of the Kawasaki dynamics accelerates the transformation process without changing the qualitative behavior. In the growth-decay regime of the Glauber dynamics, regime in which the splitting and coagulation, and creation and annihilation processes due to single-spin flips are negligible, the Kawasaki dynamics strongly increases the fraction of transformed phase because of the splitting and coagulation of clusters induced by the spin-exchange processes. Acting alone, the Kawasaki dynamics leads to the growth of the clusters of each of the phases after the quenching of the temperature to a lower value. When the final temperature T-f is smaller than a certain temperature T-f0, the average cluster radius grows linearly with time during both the initial and intermediate stages of the kinetic process, and diverges as log(2)(t(d)-t)(-1) when the time t approaches the value t(d) at which infinite clusters arise. It is shown that, among the various spin-exchange processes involved in Kawasaki dynamics, the main contribution is provided by those which decrease or increase the number of clusters by unity.(C) 2004 American Institute of Physics.