Cofiltered limits need not commute with finite colimits in sets

Let $A_n=\{m\in \mathbb{N}:m\ge n\}$ with the inverse-system inclusions $A_{n+1}\hookrightarrow A_n$, and let $B_n=C_n=1$. Each pushout of $1\leftarrow A_n\to 1$ is a singleton because $A_n$ is nonempty. Yet $\underleftarrow{\lim }{A}_n=\varnothing$, so the pushout after taking limits is $1{\bigsqcup }_{\varnothing }1$, a two-element set. Thus cofiltered limits do not in general preserve pushouts.