We study dynamics of Bragg grating solitons in a system of linearly coupled Bragg gratings with Kerr nonlinearity. The effects of dispersive reflectivity on the behaviour of solitons in the system are investigated by solving the coupled mode equations numerically. Gap solitons, are found to exist throughout the bandgap of the structure. The system supports two types of symmetric and asymmetric solitons that can have any velocities from zero to the speed of light in the medium. At given soliton parameters a critical coupling coefficient is found above which only symmetric solitons exist. Below the critical point however, both types of gap solitons may exist at the same time. Linear forms of coupled mode equations are solved analytically. The results are in excellent agreement with the gap soliton tails. Also, using the linear analysis a condition is found for the solitons to have sidelobes in their tails. Stability of solitons are investigated using systematic simulations. Generally, when dispersive reflectivity is zero, asymmetric solitons are stable for ω≥0. While with increase of dispersive reflectivity the stable region expands into the negative frequencies. Symmetric solitons on the other hand are found to be stable where they exist on their own. Interactions of quiescent gap solitons in the model are studied numerically. The outcomes generally depend on the initial separation (Δx) and phase difference. However, when the dispersive reflectivity is small, Δx-dependence is very weak. Interactions are found to result in a number of outcomes including merger into a single quiescent soliton, destruction, formation of a bound state that eventually breaks up into two separating solitons, formation of two moving and one quiescent solitons, and repulsion. The most interesting outcomes of the collisions of counter-propagating in-phase moving solitons are merger and 2→3 transformation. On the contrary, out-of-phase collisions generally result in the repulsion of the pulses.