Continuity method with movable singularities for classical Monge-Amp\`ere equations

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 $\{\omega \}$.