We develop a new $L^1$ law of large numbers where the $i$-th summand is given by a function $h(\cdot)$ evaluated at $X_i - θ_n$, and where $θ_n \circeq θ_n(X_1,X_2,\ldots,X_n)$ is an estimator converging in probability to some parameter $θ\in \mathbb{R}$. Under broad technical conditions, the convergence is shown to hold uniformly in the set of estimators interpolating between $θ$ and another consistent estimator $θ_n^{\star}$. Our main contribution is the treatment of the case where $|h|$ blows up at $0$, which is not covered by standard uniform laws of large numbers.