Starting with proposals by Schuster and Toro (2013), the massless "continuous-spin" or "infinite-helicity" irreducible representations of the Poincaré group have been the subject of several recent investigations. Their status as string-localizable particles was shown in principle by Mund, Schroer and Yngvason (2006). From the viewpoint of Wigner's equations of motion for such (as yet unobserved) particles, we endeavour to give a first-quantized approach to them, based on a background of classical elementary systems.