Context. Core collapse supernovae (CCSNe) are important sources of interstellar dust, which are potentially capable of producing 1 M of dust in their explosively expelled ejecta. However, unlike other dust sources, the dust has to survive the passage of the reverse shock, generated by the interaction of the supernova blast wave with its surrounding medium. Knowledge of the net amount of dust produced by CCSNe is crucial for understanding the origin and evolution of dust in the local and high-redshift Universe. Aims. We identify the dust destruction mechanisms in the ejecta and derive the net amount of dust that survives the passage of the reverse shock. Methods. We use analytical models for the evolution of a supernova blast wave and of the reverse shock with special application to the clumpy ejecta of the remnant of Cassiopeia A (Cas A). We assume that the dust resides in cool oxygen-rich clumps, which are uniformly distributed within the remnant and surrounded by a hot X-ray emitting plasma (smooth ejecta), and that the dust consists of silicates (MgSiO 3 ) and amorphous carbon grains. The passage of the reverse shock through the clumps gives rise to a relative gas-grain motion and also destroys the clumps. While residing in the ejecta clouds, dust is processed via kinetic sputtering, which is terminated either when the grains escape the clumps or when the clumps are destroyed by the reverse shock. In either case, grain destruction proceeds thereafter by thermal sputtering in the hot shocked smooth ejecta. Results. We find that 11.8 and 15.9 percent of silicate and carbon dust, respectively, survive the passage of the reverse shock by the time the shock has reached the centre of the remnant. These fractions depend on the morphology of the ejecta and the medium into which the remnant is expanding, as well as the composition and size distribution of the grains that formed in the ejecta. Results will therefore differ for different types of supernovae.