  We investigate the dynamics of the quantum Ising model on two-dimensional
square lattices up to $16 \times 16$ spins. In the ordered phase, the model is
predicted to exhibit dynamically constrained dynamics, leading to confinement
of elementary excitations and slow thermalization. After demonstrating the
signatures of confinement, we probe the dynamics of interfaces in the
constrained regime through sudden quenches of product states with domains of
opposite magnetization. We find that the nature of excitations can be captured
by perturbation theory throughout the confining regime, and identify the
crossover to the deconfining regime. We systematically explore the effect of
the transverse field on the modes propagating along flat interfaces and
investigate the crossover from resonant to diffusive melting of a square of
flipped spins embedded in a larger lattice.
