Let $B$ be a ring which is the colimit of rings $B_\lambda$.  Let $X_\lambda$ be a stack (not necessarily algebraic) over $B_\lambda$ such that $X_\lambda \times_{B_\lambda} B_\mu = X_\mu$ and let $X = X_\lambda \times_{B_\lambda} B$.

If $X$ is an algebraic stack, then does some $X_\lambda$ have to be algebraic?  Are there assumptions we can add to make this true?  What if the $X_\lambda$ are sheaves (so that the question becomes: if $X$ is an algebraic space, then is some $X_\lambda$ an algebraic space)?