Suppose we are given two smooth projective varieties $X$ and $Y$ over $k$. Let $D^b(X)$ and $D^b(Y)$ denote the derived category of coherent sheaves. Furthermore let $k\subset L$ be a finite Galois extension of $k$. Suppose there is an equivalence $F:D^b(X\otimes_k L)\rightarrow D^b(Y\otimes_k L)$.

If one has an object $\mathcal{T}$ in $D^b(X\otimes_k L)$ that descents i.e. there is an object $\mathcal{K}$ in $D^b(X)$ such that $\pi^*\mathcal{K}\simeq \mathcal{T}$, where $\pi:X\otimes_k L\rightarrow X$ is the projection, does the object $F(\mathcal{T})$ also descent?

Maybe this depends on the functor $F$ ?