Riverbrain and AW are each referencing gauges on two different stretches. The gauge that AW references for all sections of the Eagle River is the Eagle River near Minturn gauge. This gauge is located near the Gilman takeout, at Tigawon Road/Hwy 24. It is upstream of the mouths of Cross Creek, Two Elk Creek, Gore Creek, and a few other small creeks that flow into the Eagle between this gauge and the put-ins between Minturn and Avon. If you are putting in downstream of Minturn, I would use the Eagle River Blw Wastewater Treatment Plant at Avon gauge. This is the one that Riverbrain references. This gauge is reflective of the flow that you will see on the stretch of river from Minturn to Avon. For the stretch below Edwards to Eagle, I would look at the Eagle River Below Milk Creek near Wolcott gauge, as this includes additional water from Lake Creek, Alkali Creek and Milk Creek. So, for most rivers (except the Poudre, Colorado at Grand Junction, and maybe a few others with diversions) the flow will increase as you move downstream, because tributaries join the main river and add to the flow.
USGS Current Conditions for USGS 09067020 EAGLE R BLW WASTEWATER TREATMENT PLANT AT AVON, CO