Kleisli and colimits

Does anyone know if the Kleisli construction (sending a monad to its Kleisli category) behaves in any decent way with respect to colimits? E.g. does it in any sense preserve or reflect them? The actual situation that I have is a fixed category C, and a certain coequalizer diagram in the category of monads on C. The resulting fork in Cat is also a coequalizer, and the proofs that both diagrams are coequalizers have some ingredients in common, but I can't at present see how to deduce one from the other. Thanks, Tom Leinster