Is my generalization of uniform tightness for random variables mentioned in mathematical literature?

Question

We consider a probability space $(\Omega,\mathcal{F},\mathbb{P})$, a sequence $(X_{n})_{n\in\mathbb{N}}$ of random variables on that space and a sequence $(f_{m})_{m\in\mathbb{N}}$ of real functions. The set $\{f_{m}(X_{n})|m,n\in\mathbb{N}\}$ is uniformly tight in $m$ and $n$ (w.r.t. $(\Omega,\mathcal{F},\mathbb{P})$) if \begin{eqnarray*} \forall\epsilon>0\exists K>0\forall m,n\in\mathbb{N}:\quad\mathbb{P}(|f_{m}(X_{n})|>K)\leq\epsilon. \end{eqnarray*} I am currently working on the following more general property: \begin{eqnarray*} \forall\epsilon>0\exists K>0\exists N\in\mathbb{N}\forall m\in\mathbb{N}:\quad\sup_{n\geq N}\mathbb{P}(|f_{m}(X_{n})|>K)\leq\epsilon. \end{eqnarray*} Does this generalization (or similar properties) exist in mathematical literature? Would "semiuniform tightness" be an appropriate name for it?