  We present a theoretical investigation of the chromatic dynamics of the
witness beam within a plasma based accelerator. We derive the single particle
motion of an electron in an ion column within a nonlinear, blowout wake
including adiabatic dampening and adiabatic variations in plasma density. Using
this, we calculate the evolution of the beam moments and emittance for an
electron beam. Our model can handle near arbitrary longitudinal phase space
distributions. We include the effects of energy change in the beam, imperfect
wake loading, initial transverse offsets of the beam, and mismatch between the
beam and plasma. We use our model to derive analytic saturation lengths for the
projected, longitudinal slice, and energy slice emittance under different beam
loading conditions. Further, we show that the centroid oscillations and spot
sizes vary between the slices and the variation depends strongly on the beam
loading. Next, we show how a beam evolves in a full plasma source with density
ramps and show that the integral of the plasma density along the ramp
determines the impact on the beam. Finally, we derive several simple scaling
laws that show how to design a plasma based injector to produce a target beam
energy and energy spread.
