Motivated by the growing demand for serving large language model inference requests, we study distributed load balancing for global serving systems with network latencies. We consider a fluid model in which continuous flows of requests arrive at different frontends and need to be routed to distant backends for processing whose processing rates are workload dependent. Network latencies can lead to long travel times for requests and delayed feedback from backends. The objective is to minimize the average latency of requests, composed of the network latency and the serving latency at the backends. We introduce Distributed Gradient Descent Load Balancing (DGD-LB), a probabilistic routing algorithm in which each frontend adjusts the routing probabilities dynamically using gradient descent. Our algorithm is distributed: there is no coordination between frontends, except by observing the delayed impact other frontends have on shared backends. The algorithm uses an approximate gradient that measures the marginal impact of an additional request evaluated at a delayed system state. Equilibrium points of our algorithm minimize the centralized optimal average latencies, and we provide a novel local stability analysis showing that our algorithm converges to an optimal solution when started sufficiently close to that point. Moreover, we present sufficient conditions on the step-size of gradient descent that guarantee convergence in the presence of network latencies. Numerical experiments show that our algorithm is globally stable and optimal, confirm our stability conditions are nearly tight, and demonstrate that DGD-LB can lead to substantial gains relative to other load balancers studied in the literature when network latencies are large.