Will downstream routers update over the Zigbee net?

ZHA does support queing updates so should not update all at one (Zigbee2MQTT does not yet do that).

However I find best is to only update a single Zigbee device at a time if absolutely need to update them.

But likely you first also need to optimize your setup and network to avoid EMF/EMI/RMI interference and maybe also add more Zigbee Router devices for all the data to arrive in time before time out, see Zigbee networks: how to guide for avoiding interference + optimizing using Zigbee Router devices (repeaters/extenders) to get best possible range and coverage

That is really the best preactice for Zigbee devices. It the device is not broken then do not try fixing it :wink: