For the sake of simplicity, let's 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.