The time‐dependent behavior of a transversely magnetized, two‐dimensional plasma–wall sheath has been studied through particle simulations, with the aim of modeling plasma behavior in the vicinity of the limiters and walls of magnetized plasma devices. The model assumes a magnetic field perfectly parallel to the confining surfaces. The simulations have shown that the cross‐field sheath between a wall and a plasma is a self‐sustaining turbulent boundary layer, with strong potential fluctuations and anomalous particle transport. The driving mechanism for this turbulence is the Kelvin–Helmholtz instability, which arises from the sheared particle drifts created near the wall. In this paper, the transient behavior leading to the turbulent steady state is presented, and the processes of linear growth, vortex saturation, and vortex coalescence are examined. An analytic model for the boundary Kelvin–Helmholtz instability is derived and shown to correctly predict the growth rates of the long‐wavelength modes. In a companion paper, the steady‐state structure and behavior of the cross‐field sheath will be discussed in detail.