Adaptive Systems-on-Chips (SoCs) are increasingly being used in mixed criticality systems (MCSs), such as in autonomous driving, aviation and medical systems. In this context, AMD has proposed the Versal SoC, which has a heterogeneous architecture including, among other components, an Artificial Intelligence Engine (AIE), which is a 2D array of processors and memory tiles designed for AI and signal processing workloads. While this AIE offers significant potential for accelerating real-time data processing tasks, this has not yet been explored in the context of MCSs since individual tasks with different criticality levels cannot be dynamically assigned to tiles due to the static mapping of dataflow graphs and tasks. In this work, we propose a dynamic task dispatching infrastructure that enables task switching on the AIE at runtime. Based on this infrastructure, we present an MCS design that dynamically assigns tasks of different criticality to a pool of AIE tiles, depending on the criticality mode of the system. Our approach overcomes the limitations of static dataflow graph mappings and, for the first time, exploits the parallel processing capabilities of the AIE for MCSs. We also present a comprehensive timing analysis of the overhead introduced by the task dispatcher infrastructure, focusing on control logic, context switching and data copy operations. This shows that these operations have low variance and are negligible compared to the overall execution time, demonstrating that our infrastructure is suitable for MCSs. Finally, we evaluate the proposed infrastructure using an autonomous driving workload with tasks that have variable execution times and different criticality levels. In this case study, we maximized AIE utilization, reducing idle time by 65.5 %, while measuring an execution time overhead of less than 0.002 %, and doubling the throughput of low-criticality tasks.