We introduce consecutive-pattern-avoiding stack-sorting maps $\text{SC}_σ$, which are natural generalizations of West's stack-sorting map $s$ and natural analogues of the classical-pattern-avoiding stack-sorting maps $s_σ$ recently introduced by Cerbai, Claesson, and Ferrari. We characterize the patterns $σ$ such that $\text{Sort}(\text{SC}_σ)$, the set of permutations that are sortable via the map $s\circ\text{SC}_σ$, is a permutation class, and we enumerate the sets $\text{Sort}(\text{SC}_σ)$ for $σ\in\{123,132,321\}$. We also study the maps $\text{SC}_σ$ from a dynamical point of view, characterizing the periodic points of $\text{SC}_σ$ for all $σ\in S_3$ and computing $\max_{π\in S_n}|\text{SC}_σ^{-1}(π)|$ for all $σ\in\{132,213,231,312\}$. In addition, we characterize the periodic points of the classical-pattern-avoiding stack-sorting map $s_{132}$, and we show that the maximum number of iterations of $s_{132}$ needed to send a permutation in $S_n$ to a periodic point is $n-1$. The paper ends with numerous open problems and conjectures.