Which portion of a river should have the highest velocity? Upper course (gradient) or lower course (smooth bank), or can't this question be generalized?

Velocity is highest in the lower course owing to:

River channel is smooth, materials forming the bed have been reduced to fine particles therefore less friction.

Hydraulic pressure from the water upstream 'pushes' the water along.

The flow of water is laminar (like of sliding pack of cards) which is more efficient than the turbulent flow in the upper sections of a river.

http://water.usgs.gov/edu/measureflow.html

However, waterfalls are more likely to be in the middle course, and obviously velocity will be highest there.

I've got a site here that goes the other way - highest velocity is in the upper course where the gradient is steeper and the course straighter. In the lower courses of many rivers, meanders slow down the flow. In terms of volume, of course the lower river carries more water, but it does it far slower. I'm not quite sure if waterfalls (of the Niagara type) qualify for this question, as they are not flowing between banks.

http://www.scienceclarified.com/landforms/Ocean-Basins-to-Volcanoes/Stream-and-River.html

The cross-section of the riverbed seems to have an effect. A shallow wide stream will not flow as fast as fast as a squarer section one (or preferably a U section to avoid turbulence in the corners). Drag is caused by the volume of silt suspended in lower courses (esp. Mississippi), and this tends to slow the flow. Roughness in a streambed will also cause some reduction of flow speed, but obstructing rocks also have the effect of forcing the water into narrower (therefore faster) channels. Open to argument both ways.
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