It can be either depending on the specific type of servomotor (variable reluctance, permanent magnet, or hybrid). Tuning of the PID is important.
Particularly because some of these motor technologies can have forbidden speed ranges, within which they have zero output torque. It is important that whatever speed you operate at and whatever be your controller tuning, the motor should not pass through such speeds during either acceleration or deceleration. To that extent, constant speed operation is marginally convenient than variable speeds. But both modes are possible.
Great answer by Sanjoy. The attached article might be of interest http://www.ni.com/white-paper/3782/en/ but my answer would say it depends on the location research or in production. Production needs the simpler solution to survive.
When implementing a PID controller for a continuous rotation servo, you can specify a constant speed or a variable speed depending on the requirements. For high demanding systems three PID control loops are implemented for current, speed, and position. You can study some advanced control strategies for a in the paper “A New Adaptive Self-Tuning Fourier Coefficients Algorithm for Periodic Torque Ripple Minimization in Permanent Magnet Synchronous Motors (PMSM)” at https://www.researchgate.net/profile/Alfonso_Gomez-Espinosa/contributions