You need a continuity condition on the cocycles, otherwise it is probably false. The coherent sheaf automatically has a model over a finite Galois extension $L$ of $K$ contained in the fixed algebraic closure, and the continuity condition tells you that you can choose $L$ so that the cocycle factors through $Gal(L/K)$. Now apply finite descent.

You need a continuity condition on the cocycles, otherwise it is probably false. The coherent sheaf automatically has a model over a finite Galois extension $L$ of $K$ contained in the fixed algebraic closure, and the continuity condition tells you that you can choose $L$ so that the cocycle factors through $Gal(L/K)$. Now apply finite descent.

Added: The continuity condition just says that the descent datum splits over a finitely generated field extension of the base field. There is an elementary discussion of such things here

You need a continuity condition on the cocycles, otherwise it is probably false. The quasi-coherent sheaf automatically has a model over a finite Galois extension $L$ of $K$ contained in the fixed algebraic closure, and the continuity condition tells you that you can choose $L$ so that the cocycle factors through $Gal(L/K)$. Now apply finite descent.

You need a continuity condition on the cocycles, otherwise it is probably false. The coherent sheaf automatically has a model over a finite Galois extension $L$ of $K$ contained in the fixed algebraic closure, and the continuity condition tells you that you can choose $L$ so that the cocycle factors through $Gal(L/K)$. Now apply finite descent.