  On a compact K\"ahler manifold $(X,\omega)$, we study the strong continuity
of solutions with prescribed singularities of complex Monge-Amp\`ere equations
with integrable Lebesgue densities. Moreover, we give sufficient conditions for
the strong continuity of solutions when the right-hand sides are modified to
include all (log) K\"ahler-Einstein metrics with prescribed singularities. Our
findings can be interpreted as closedness of new continuity methods in which
the densities vary together with the prescribed singularities. For
Monge-Amp\`ere equations of Fano type, we also prove an openness result when
the singularities decrease. As an application, we deduce a strong stability
result for (log-)K\"ahler Einstein metrics on semi-K\"ahler classes given as
modifications of $\{\omega\}$.
