For the sake of simplicity, let us assume $X=nX_{\textrm{red}}$, where $X_{\textrm{red}}$ is a reduced scheme over $\mathbb{C}$ and $n \geq 2$.
Let $\Delta$ be a small disk and let $p_n \colon \Delta \to \Delta$ be the map given by $p_n(z)=nz$.
Let $\pi \colon X_{\textrm{red}} \times \Delta \to \Delta$ be the natural projection. Therefore $q:=p_n \circ \pi$ is the family you are looking for.
Indeed, the fibre of $q$ over the point $t\neq 0$ consists of $n$ disjoint copies of $X_{\textrm{red}}$, whereas the fibre over $0$ is $X$.
Of course, you can also take $\textrm{Spec }k[x]$ or $\textrm{Spec }k[[x]]$ instead of $\Delta$, if you prefer so.