The transition sequence in the Heusler alloy Ni50Mn34In8Ga8 has been determined from measurements of elasticity, heat flow and magnetism to be paramagnetic austenite → paramagnetic martensite → ferromagnetic martensite at ∼335 and ∼260 K, respectively, during cooling. The overall pattern of elastic stiffening/softening and acoustic loss is typical of a system with bilinear coupling between symmetry breaking strain and the driving structural/electronic order parameter, and a temperature interval below the transition point in which ferroelastic twin walls remain mobile under the influence of external stress. Divergence between zero-field-cooling and field-cooling determinations of DC magnetisation below ∼220 K indicates that a frustrated magnetic glass develops in the ferromagnetic martensite. An AC magnetic anomaly which shows Vogel-Fulcher dynamics in the vicinity of ∼160 K is evidence of a further glassy freezing process. This coincides with an acoustic loss peak and slight elastic stiffening that is typical of the outcome of freezing of ferroelastic twin walls. The results suggest that local strain variations associated with the ferroelastic twin walls couple with local moments to induce glassy magnetic behaviour.