Plasma-based accelerators offer high accelerating gradients and scalability through staging or long plasma sources, which makes them good candidates for future accelerator and collider concepts. Proton-driven accelerators in particular have the potential to bring particles to high energy in a single stage. In the quasilinear regime, where the plasma wake is only partially evacuated, a witness bunch of electrons drives a cavitated wake, which acts to preserve the emittance of the portion of the witness inside this self-blowout. In the case of a misalignment between the driver and witness, this behaviour can persist, but its effectiveness is reduced. In this paper, we study transverse witness dynamics in this regime, and develop an analytical model to describe the witness motion. We show that the degree of emittance preservation can be predicted by estimating the beam density after phase mixing. Particle in cell simulations using the AWAKE Run 2c baseline parameters show excellent agreement with the predictive model developed. This work allows alignment constraints to be set both for the AWAKE experiment and other wakefield acceleration schemes operating in the quasilinear regime.