Consider a sequence of functions $\{ f_n\}_{n \geq 1}$, where each $f_n : X \to \mathbb{R}$ for a metric space $X$. Let $r_n$ be a sequence of positive reals diverging to $+\infty$, and suppose that for all $\epsilon, \delta > 0$, there exists $N$ such that $|f_n(x) -f_n(y)| < \epsilon$ for all $\| x-y\|_X < \delta / r_n$ and $n \geq N$. Equivalently, if $$\omega_n(\delta) := \sup\{ | f_n(x) - f_n(y)| : x,y \in X, \|x - y\|_X \leq \delta\}$$ is the modulus of continuity for $f_n$ for each $n\geq 1$, then $\omega_n(\delta / r_n) \to 0$ for all $\delta >0$.
This property is implied by equicontinuity of $\{f_n\}_{n \geq 1}$, since $\delta / r_n \to 0$ for all $\delta > 0$. However, it is strictly weaker than equicontinuity. For example, if each $f_n$ is differentiable and $\sup_{x \in X} |f_n'(x)| = t_n < \infty$ such that $t_n \to \infty$, then $\{f_n\}_{n \geq 1}$ is not (may not be?) equicontinuous but the above property holds for any sequence $r_n$ such that $t_n / r_n \to 0$.
Does anyone know if this property has a name?