  We consider the KP-I and gKP-I equations in $\mathbb{R}\times
(\mathbb{R}/2\pi \mathbb{Z})$. We prove that the KdV soliton with subcritical
speed $0<c<c^*$ is orbitally stable under the global KP-I flow constructed by
Ionescu and Kenig \cite{IK}. For supercritical speeds $c>c^*$, in the spirit of
the work by Duyckaerts and Merle \cite{DM}, we sharpen our previous instability
result and construct a global solution which is different from the solitary
wave and its translates and which converges to the solitary wave as time goes
to infinity. This last result also holds for the gKP-I equation.
