 Global-section base-change failure in a flat projective family

Flatness of a projective family does not by itself force global sections to commute with base change. For a discrete valuation ring with uniformizer $\pi$, glue $\mathrm{Spec}R[\pi t,t^2,t^3]$ to $\mathrm{Spec}R[t^{-1}]$ along $\mathrm{Spec}R[t,t^{-1}]$. In the projective model with coordinates $(1,\pi t,t^2,t^3)$, the global ring is $R$. The special-fibre first chart has an additional square-zero element $a$ annihilated by $t^2,t^3$; it vanishes on the overlap and therefore extends by zero to a global section. The special-fibre global ring is $k[a]/(a^2)$, and base change from $R$ reaches only its constants.