We study the problem in which a central planner sequentially allocates a single resource to multiple strategic agents using their utility reports at each round, but without using any monetary transfers. We consider general agent utility distributions and two standard settings: a finite horizon $T$ and an infinite horizon with $\gamma$ discounts. We provide general tools to characterize the convergence rate between the optimal mechanism for the central planner and the first-best allocation if true agent utilities were available. This heavily depends on the utility distributions, yielding rates anywhere between $1/\sqrt T$ and $1/T$ for the finite-horizon setting, and rates faster than $\sqrt{1-\gamma}$, including exponential rates for the infinite-horizon setting as agents are more patient $\gamma\to 1$. On the algorithmic side, we design mechanisms based on the promised-utility framework to achieve these rates and leverage structure on the utility distributions. Intuitively, the more flexibility the central planner has to reward or penalize any agent while incurring little social welfare cost, the faster the convergence rate. In particular, discrete utility distributions typically yield the slower rates $1/\sqrt T$ and $\sqrt{1-\gamma}$, while smooth distributions with density typically yield faster rates $1/T$ (up to logarithmic factors) and $1-\gamma$.