An expression for the rate-of-strain tensor on a rigid surface due to Caswell is generalized to an arbitrarily moving and continuously deforming surface or interface between two immiscible fluids. Corresponding expressions for the velocity gradient and vorticity tensors are derived in an inertial frame of reference. A noteworthy feature of the expression for the rate-of-strain tensor is the presence of a tangent-tangent component, which is absent in the case of a rigid surface. Kinematic applications based on numerical solutions of the Navier–Stokes equation for laminar and turbulent flow demonstrate the significance and implications of the derived expressions.