[joneholland]

The client doesn't do it. You put your front ends behind a load balancer like an ELB, or use a reverse proxy like Nginx.

Edit: And yes, round robin is the most commonly used load distribution technique, and works very well assuming each request has a roughly equivalent unit of work cost.

[hyperpape]

I'm surprised you wouldn't run into cases where the requests being rate-limited can't end up unevenly distributed between servers. There are assumptions you could add that would make that not a problem, but I'm surprised they'd hold.