Load balancing is an important activity in distributed systems. At a high-level of abstraction, this problem assumes processes in the system begin with a potentially uneven assignment of “work” which they must then attempt to spread evenly. There are many different approaches to solving this problem. In this paper, we focus on local load balancing — an approach in which work is balanced in an iterative and distributed manner by having processes exchange work in each round with neighbors in an underlying communication network. The goal with local load balancing is to prove that these local exchanges will lead over time to a global balance. We describe and analyze a new local load balancing strategy called max neighbor, and prove a bound on the number of rounds required for it to obtain a parameterized level of balance, with high probability, in a general dynamic network topology. We then prove this analysis tight (within logarithmic factors) by describing a network and initial work distribution for which max neighbor matches its upper bound, and then build on this to prove that no load balancing algorithm in which every node exchanges work with at most one partner per round can converge asymptotically faster than max neighbor.