  By a result of Biswas and Dos Santos, on a smooth and projective variety over
an algebraically closed field, a vector bundle trivialized by a proper and
surjective map is essentially finite, that is it corresponds to a
representation of the Nori fundamental group scheme. In this paper we obtain
similar results for non-proper non-smooth algebraic stacks over arbitrary
fields of characteristic $p>0$. As by-product we have the following partial
generalization of the Biswas-Dos Santos' result in positive characteristic: on
a pseudo-proper and inflexible stack of finite type over $k$ a vector bundle
which is trivialized by a proper and flat map is essentially finite.
