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We address the stability analysis of composite hybrid dynamical feedback systems of the type depicted in Fig. 1, consisting of a block (usually the plant) which is described by an operator and of a finite-dimensional block described by a system of difference equations (usually a digital controller). We establish results for the well-posedness, attractivity, asymptotic stability, uniform boundedness, asymptotic stability in the large, and exponential stability in the large for such systems. The hypotheses of our results are phrased in terms of the I/O properties of and in terms of the Lyapunov stability properties of the subsystem described by the indicated difference equations. The applicability of our results is demonstrated by two specific examples.