In safety-critical systems many software components of different criticalities or assurance levels need to interact in a timely manner to keep the system and environment safe. Nowadays, these systems are challenged by technological progress resulting in rapid increases in both software complexity and processing demands. Efficiently designing safety-critical systems subject to stringent timing requirements is therefore a challenge and a necessity. In this article, we consider the mixed-criticality execution model and homogeneous multi-core processors. We begin by defining a task model incorporating mixed-criticality, real-time and precedence constraints in the form of directed acyclic graphs. A meta-heuristic to solve the scheduling problem of this task model is then defined and proved to respect deadlines, even when the system needs to give more processing power to the most critical tasks. The state-of-the-art techniques capable of scheduling a similar task model have only been developed for dual-criticality systems. Conversely, the meta-heuristic we propose has been generalized to support an arbitrary number of criticality levels. We instantiated our meta-heuristic adopting scheduling algorithms such as G-EDF, G-LLF, or G-EDZL for each level of criticality. The experiments show excellent results in terms of acceptance ratio and number of preemptions.