Let $R$ be the excess over the boundary in renewal theory. It is well known that $ER$ has a limit $r$ when the drift of the random walk $\mu \geq 0$. We study renewal theorems with varying $\mu$. Conditions are given under which the tail $ER - r$ is uniformly dominated by a decreasing integrable function for $\mu$ in a compact interval in $(0, \infty)$. Conditions are also given under which the derivative of the tail $(\partial/\partial\mu)(ER - r)$ is uniformly dominated by a directly Riemann integrable function.
"A Renewal Theory with Varying Drift." Ann. Probab. 17 (2) 723 - 736, April, 1989. https://doi.org/10.1214/aop/1176991423