Core-collapse supernova remnants are structures of the interstellar medium (ISM) left behind the explosive death of most massive stars ($\lesssim 40\, \rm {\rm M}_{\odot }$). Since they result in the expansion of the supernova shock wave into the gaseous environment shaped by the star’s wind history, their morphology constitutes an insight into the past evolution of their progenitor star. Particularly, fast-moving massive stars can produce asymmetric core-collapse supernova remnants. We investigate the mixing of materials in core-collapse supernova remnants generated by a moving massive $35\, \rm {\rm M}_{\odot }$ star, in a magnetised ISM. Stellar rotation and the wind magnetic field are time-dependently included into the models which follow the entire evolution of the stellar surroundings from the zero-age main-sequence to $80\, \rm kyr$ after the supernova explosion. It is found that very little main-sequence material is present in remnants from moving stars, that the Wolf-Rayet wind mixes very efficiently within the $10\, \rm kyr$ after the explosion, while the red supergiant material is still unmixed by 30% within $50\, \rm kyr$ after the supernova. Our results indicate that the faster the stellar motion, the more complex the internal organisation of the supernova remnant and the more effective the mixing of ejecta therein. In contrast, the mixing of stellar wind material is only weakly affected by progenitor motion, if at all.