It is well-known that the general p-curvature conjecture for algebraic vector bundles with connection on arbitrary varieties reduces to the case of the thrice-punctured projective line. However, this reduction is does not respect the rank of the vector bundle. In this paper, we study the condition of vanishing p-curvatures in families of vector bundles with connection. Using this analysis, along with results from Ananth Shankar's thesis and some group-theoretic techniques, we reduce the p-curvature conjecture for rank 2 bundles on a generic curve (meaning the geometric generic fiber of the universal curve over moduli space) to the rank 2 (same rank!) p-curvature conjecture on thrice-punctured lines. This latter case was previously established by Katz, and therefore we obtain a proof for rank 2 bundles over a generic curve.