Let $R$ be a dvr with fraction field $K$ and residue field $k$. Let $\mathcal{X}\to \mathcal{Y}$ be a morphism of $R$-schemes such that $\mathcal{X}_K\to \mathcal{Y}_K$ is non-constant.

Is the morphism on special fibres $\mathcal{X}_k\to \mathcal{Y}_k$ non-constant?

The answer should be "no", because one can take a morphism $\mathbb{A}^1_R\to \mathbb{A}^1_R$ of the form $x\mapsto \pi x$ with $\pi$ the uniformiser of $R$.

My question is really:

Under what additional conditions on $\mathcal{X}$ and $\mathcal{Y}$ is the morphism $\mathcal{X}_k\to \mathcal{Y}_k$ non-constant? What if $\mathcal{X}$ and $\mathcal{Y}$ are projective over $R$?