Let B$B$ be a ring which is the colimit of rings B_\lambda$B_\lambda$. Let X_\lambda$X_\lambda$ be a stack (not necessarily algebraic) over B_\lambda$B_\lambda$ such that X_\lambda \times_{B_\lambda} B_\mu = X_\mu$X_\lambda \times_{B_\lambda} B_\mu = X_\mu$ and let X = X_\lambda \times_{B_\lambda} B$X = X_\lambda \times_{B_\lambda} B$.
If X$X$ is an algebraic stack, then does some X_\lambda$X_\lambda$ have to be algebraic? Are there assumptions we can add to make this true? What if the X_\lambda$X_\lambda$ are sheaves (so that the question becomes: if X$X$ is an algebraic space, then is some X_\lambda$X_\lambda$ an algebraic space)?
