  In this paper, we study the minimal model theory for threefolds in mixed
characteristic. As a generalization of a result of Kawamata, we show that the
MMP holds for strictly semi-stable schemes over an excellent Dedekind scheme
$V$ of relative dimension two without any assumption on the residue
characteristics of $V$. We also prove that we can run a $(K_{X/V}+\Delta)$-MMP
over $Z$, where $\pi \colon X \to Z$ is a projective birational morphism of
$\mathbb{Q}$-factorial quasi-projective $V$-schemes and $(X,\Delta)$ is a
three-dimensional dlt pair with $\mathrm{Exc}(\pi) \subset \lfloor \Delta
\rfloor $.
