  The standard dynamical approach to quantum thermodynamics is based on
Markovian master equations describing the thermalization of a system weakly
coupled to a large environment, and on tools such as entropy production
relations. Here we develop a new framework overcoming the limitations that the
current dynamical and information theory approaches encounter when applied to
this setting. More precisely, we introduce the notion of continuous
thermomajorization, and employ it to obtain necessary and sufficient conditions
for the existence of a Markovian thermal process transforming between given
initial and final energy distributions of the system. These lead to a complete
set of generalized entropy production inequalities including the standard one
as a special case. Importantly, these conditions can be reduced to a finitely
verifiable set of constraints governing non-equilibrium transformations under
master equations. What is more, the framework is also constructive, i.e., it
returns explicit protocols realizing any allowed transformation. These
protocols use as building blocks elementary thermalizations, which we prove to
be universal controls. Finally, we also present an algorithm constructing the
full set of energy distributions achievable from a given initial state via
Markovian thermal processes and provide a $\texttt{Mathematica}$ implementation
solving $d=6$ on a laptop computer in minutes.
