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)?