  Master equations are a useful tool to describe the evolution of open quantum
systems. In order to characterize the mathematical features and the physical
origin of the dynamics, it is often useful to consider different kinds of
master equations for the same system. Here, we derive an exact connection
between the time-local and the integro-differential descriptions, focusing on
the class of commutative dynamics. The use of the damping-basis formalism
allows us to devise a general procedure to go from one master equation to the
other and vice-versa, by working with functions of time and their Laplace
transforms only. We further analyze the Lindbladian form of the time-local and
the integro-differential master equations, where we account for the appearance
of different sets of Lindbladian operators. In addition, we investigate a
Redfield-like approximation, that transforms the exact integro-differential
equation into a time-local one by means of a coarse graining in time. Besides
relating the structure of the resulting master equation to those associated
with the exact dynamics, we study the effects of the approximation on
Markovianity. In particular, we show that, against expectation, the coarse
graining in time can possibly introduce memory effects, leading to a violation
of a divisibility property of the dynamics.
