The question is in the title. I know that, e.g., W bosons change the flavour in electroweak processes. What do gluons do?
Emitting or absorbing a gluon changes the color of a quark. This is because gluons themselves have color.
However, as @dmckee points out, not all gauge bosons change some property (other than energy and momentum) of the particle emitting or absorbing them. For example, photons are not charged, so emitting or absorbing a photon does not change the charge of an electron.
The fact that gluons carry color while photons do not carry charge is related to the fact that the $SU(3)$ gauge symmetry group of the strong interaction is non-commutative while the $U(1)$ gauge symmetry group of the electromagnetic interaction is commutative.