Paul,Obviously there is more than one factor at play here.
There are two primary considerations going on here.
1. There is a positive acceleration force on the fly leg due to the tension at the top of the loop that is due to the momentum change of the moving mass going around the loop. That positive force is nominally proportional to the linear mass density of the line and the square of the loop velocity over the ground (assuming a tethered cast).
2. There is a negative acceleration force on the fly leg due to the drag losses on the loop, trailing line, and fly. The skin drag depends on the diameter squared of the trailing line, the square of its velocity (=2*loop velocity) and the length of the fly leg. There is also a form drag component on the fly that depends on the diameter of the fly and the square of its velocity. If the fly leg tilts down (the general case) then it will also have a sizeable form drag component that depends on sin(angle of the tilt), line diameter, velocity squared, and the length of the fly leg.
At the start of the cast the drag losses are large due to the length of the line. In the case where the fly leg is very long the combined drag losses will in general be larger than the positive tension force and the fly velocity will tend to decelerate at the beginning of the loop propagation.
Near the end of the cast the drag losses on the line are much smaller and the loop will tend to accelerate. That is especially true for a level line that has a constant linear mass density over its entire length.
However, if the end of the line is tapered then that will reduce the positive acceleration force due to the reduction in the linear mass density of the line. The linear mass density(rho_l) depends on the square of the diameter so halving the line diameter will cause a factor of 4 reduction in rho_l.
If that taper is enough then the positive acceleration force will be smaller than the drag force (at the end of the cast the drag forces on the fly may be the dominate drag loss) and the fly velocity will decrease. That is especially true when a long leader starts going around the loop. The thin leader will have a much smaller rho_l that pretty much guarantees the fly velocity will decrease and produce a smoother turn over of the fly. This is what Merlin is referring to when he talks about exponential deceleration
It takes a complicated ODE to keep track of the competing acceleration and deceleration forces as the fly propagates even for a level line. When the line is tapered it becomes a programming nightmare, and my hat is off to Lingard, Spolek, and Gatti-Bono for figuring it out.