The mobility analysis of mechanisms rests on an adequate formulation of the constraints defining its configuration space (c-space). Whereas there is no general method for a global analysis, the higher-order mobility analysis, which locally approximates the c-space, is applicable to general mechanisms. It requires an efficient method for the evaluation of higher-order constraints, i.e. constraints on velocity, acceleration, jerk, etc. Such a method is known for linkages comprising lower pair joints only. In this paper a method for the efficient evaluation of higher-order constraints for mechanisms comprising higher pair joints is proposed. The method builds on the results for the lower pair linkages. It leads to a computationally simply recursive algorithm. This is applied to the mobility analysis that allows to determine the local finite mobility, to detect singularities, and to identify shaky mechanisms.