Given an algebraically closed field $K$, a dynamical sequence over $K$ is a $K$-valued sequence of the form $a(n):= f(φ^n(x_0))$, where $φ\colon X\to X$ and $f\colon X\to\mathbb{A}^1$ are rational maps defined over $K$, and $x_0\in X$ is a point whose forward orbit avoids the indeterminacy loci of $\varphi$ and $f$. Many classical sequences from number theory and algebraic combinatorics fall under this dynamical framework, and we show that the class of dynamical sequences enjoys numerous closure properties and encompasses all elliptic divisibility sequences, all Somos sequences, and all $C^n$- and $D^n$-finite sequences for all $n\ge 1$, as defined by Jiménez-Pastor, Nuspl, and Pillwein. We also give an algorithm for proving that two dynamical sequences are identical and illustrate how to use this algorithm by showing how to prove several classical combinatorial identities via this method.