The lattice Boltzmann method is used to simulate the temporal and spatial evolution of two-dimensional oscillatory flow inside a channel. The flow is seeded with infinitesimal disturbances so as to trigger primary instabilities and subsequent turbulent bursts in the transitional flow. Certain statistics of the evolving fluctuating velocity field which are useful for understanding the transition to turbulence, such as the turbulent kinetic energy, are presented. Particles migrate in the spanwise direction in response to the fluid motion of a turbulent burst. Lagrangian trajectories of these fluid particles are used to study the particle migration in the spanwise direction. It is observed that the line least likely to be crossed by particles starting inside the Stokes boundary layer is the line marking the position of the edge of the boundary-layer itself.