The rapid growth of AI-enabled Internet of Vehicles (IoV) calls for efficient Machine Learning (ML) solutions that can handle high vehicular mobility and decentralized data. This has motivated the emergence of Hierarchical Federated Learning over vehicle-edge-cloud architectures (VEC-HFL). Nevertheless, one aspect which is underexplored in the literature on VEC-HFL is that vehicles often need to execute multiple ML tasks simultaneously, where this multi-model training environment introduces crucial challenges. First, improper aggregation rules can lead to model obsolescence and prolonged training times. Second, vehicular mobility may result in inefficient data utilization by preventing the vehicles from returning their models to the network edge. Third, achieving a balanced resource allocation across diverse tasks becomes of paramount importance as it majorly affects the effectiveness of collaborative training. We take one of the first steps towards addressing these challenges via proposing a framework for multi-model training in dynamic VEC-HFL with the goal of minimizing global training latency while ensuring balanced training across various tasks, a problem that turns out to be NP-hard. To facilitate timely model training, we introduce a hybrid synchronous-asynchronous aggregation rule. Building on this, we present a novel method called Hybrid Evolutionary And gReedy allocaTion (HEART). The framework operates in two stages: first, it achieves balanced task scheduling through a hybrid heuristic approach that combines improved Particle Swarm Optimization (PSO) and Genetic Algorithms (GA); second, it employs a low-complexity greedy algorithm to determine the training priority of assigned tasks on vehicles. Experiments on real-world datasets demonstrate the superiority of HEART over existing methods.