This paper analyses the performance of projected gradient descent on optimisation problems with cost functions and constraints that vary in discrete time. Specifically, strongly convex cost functions with Lipschitz gradient, and a sequence of convex constraints are assumed. Error bounds and suboptimality bounds are derived for a variety of cases, which show convergence to a steady-state. Conditions on the constraint sequence are also presented for guaranteeing finite-time feasibility, and for bounding the distance between successive minimisers. Numerical examples are then presented to validate the analytical results.