Transitional flows with an unsteady inflow play a vital role in a broad range of applications, including biological fluid transport to space applications. In such cases, the thickness of the boundary layer formed over the solid surface varies in both space and time, causing a high level of complexity in the path of vortical structures formed from the shear/boundary layer. Also, time and space-dependent shear stress exerted by the fluid, separation, and associated instability phenomena are to be better understood. In this work, direct numerical simulations (DNS) are performed to study the stability of vortical flow structures associated with an unsteady boundary layer under an adverse pressure gradient condition. A trapezoidal pulse of mean velocity, consisting of the acceleration phase from rest followed by the constant velocity phase and deceleration phase to rest, is imposed at the inlet of the computational domain. The impact of the spatial and temporal components on the evolution patterns of the shear-layer and three-dimensional instabilities are examined in detail. By employing dynamic mode decomposition, some key features of the transitional flow and their time dynamics are extracted.