In the present analysis, the non-local behavior during thermal lagging is studied to accommodate the effect of the thermomass for a piezoelastic half-space due to the influence of magnetic field in the context of dual-phase-lag model of generalized thermoelasticity, defined in an integral form of a common derivative on a slipping interval. Simultaneous existence of the nonlocality (in space) and lagging (in time) behaviors give rise to a new type of thermal waves, which can be of major significance. In the current analysis, Danilovskaya's problem has been solved in which, the thermal shock is employed on the boundary of the half-space, which is traction-free. Employing the Laplace transform, the problem has been solved and the solution in the space-time domain is achieved on applying Riemann-sum approximation technique. According to the graphical representations corresponding to the numerical results, conclusions about the new theory is constructed. Several graphical representations are shown to analyze the effects of different parameters and to mark the variation of this non-local model with previously existing models. Excellent predictive capability is demonstrated due to the presence of magnetic filed, memory-dependent derivative, the electric displacement and time-delay parameter, and the effect of correlating length is also reported.