Base change for semiorthogonal decompositions

Consider an algebraic variety $X$ over a base scheme $S$ and a faithful base change $T\to S$. Given an admissible subcategory $\text{CA}$ in the bounded derived category of coherent sheaves on $X$, we construct an admissible subcategory in the bounded derived category of coherent sheaves on the fiber product $X{\times }_{S}T$, called the base change of $\text{CA}$, in such a way that the following base change theorem holds: if a semiorthogonal decomposition of the bounded derived category of $X$ is given then the base changes of its components form a semiorthogonal decomposition of the bounded derived category of the fiber product. As an intermediate step we construct a compatible system of semiorthogonal decompositions of the unbounded derived category of quasicoherent sheaves on $X$ and of the category of perfect complexes on $X$. As an application we prove that the projection functors of a semiorthogonal decomposition are kernel functors.