We present a measurement of the proper motion of the presumed pulsar in the evolved composite supernova remnant (SNR) MSH 15-56 whose pulsar wind nebula (PWN) has been disrupted by the supernova (SN) reverse shock. Using Chandra X-ray observations acquired over a baseline of 15 years, we measure a pulsar velocity of 720 (+290/-215) km/s and a direction of motion of 14 +/- 22 degrees west of south. We use this measurement to constrain a hydrodynamical model for the evolution of this system and find that its morphology is well-described by an SNR expanding in an ambient density gradient that increases from east to west. The effect of the density gradient and the pulsars motion is an asymmetric interaction between the SN reverse shock and the PWN that displaces the bulk of the PWN material away from the pulsar, towards the northeast. The simulation is consistent with an SNR age of 11,000 years, an SN ejecta mass of 10 solar masses, and an average surrounding density of 0.4 cm^-3. However, a combination of a higher SN ejecta mass and ambient density can produce a similar SNR morphology at a later age.
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