Let $X\rightarrow \mathrm{Spec}\:F_q[[t]]$ be a flat morphism with smooth proper geometrically connected fibers. Suppose the central fiber lifts to a scheme $X'_0$ smooth proper over $W(F_q)$. Is our family the base change of a flat morphism with smooth proper fibers $X'\rightarrow \mathrm{Spec}\:W(F_q)[[t]]$ whose central fiber is $X'_0$? is our family the base change of any flat morphism with smooth proper fibers $X'\rightarrow \mathrm{Spec}\:W(F_q)[[t]]$?
If the answer is "no", what additional conditions are required? Is it enough to demand that the central and the generic fiber lift to smooth proper schemes in possibly unrelated ways?