Does every finite flat group scheme become constant  after finite base change? Does every finite flat group scheme $G/X$ become constant after finite base change? Which additional properties of the base change morphism can we impose?
Edit: Which conditions do we have to impose on $G/X$ so that the answer becomes "yes"?
 A: The answer to the first question is no. It is definitely possible to write down
an example directly but here is one systematic way of giving examples (in
positive characteristic).
Start with a base ring $R$, say $R=k[t]$, where $k$ is a field of positive
characteristic and an $R$-algebra $A$ which is finitely generated projective as
$R$-module. As example we take $A=R[s]/(s^2-ts)$. Then the invertible elements 
of $A$ is a smooth algebraic group scheme $A^\ast$ over $R$. This construction
commutes with base change so in our example it is $\mathbb G_m^2$ over any point
of $\mathrm{Spec}R$ different from the origon and ias $\mathbb G_m\times\mathbb
G_a$ over the origin. To get a finite group scheme we simply take the kernel of
the relative Frobenius map to get $\mu_p^2$ outside of the origin and
$\mu_p\times\alpha_p$ over the origin.
Another class of examples is to look at an abelian scheme $B$ over $R$ and look
at the kernel of multiplication on $B$ by some prime (say) $p$. Over points of
$R$ over which $p$ is invertible this kernel will have a non-trivial infinitesimal
part and over points where $p$ is invertible it will be étale. Hence we can let
$R$ be a mixed characteristic DVR with residue field characteristic $p$. We can
also let $R$ be characteristic $p$ DVR with $B$ being a supersingular elliptic
curve over the special point and ordinary over the generic.
