In recent years, there has been a growing interest in utilizing reinforcement learning (RL) to optimize long-term rewards in recommender systems. Since industrial recommender systems are typically designed as multi-stage systems, RL methods with a single agent face challenges when optimizing multiple stages simultaneously. The reason is that different stages have different observation spaces, and thus cannot be modeled by a single agent. To address this issue, we propose a novel UNidirectional-EXecution-based multi-agent Reinforcement Learning (UNEX-RL) framework to reinforce the long-term rewards in multi-stage recommender systems. We show that the unidirectional execution is a key feature of multi-stage recommender systems, bringing new challenges to the applications of multi-agent reinforcement learning (MARL), namely the observation dependency and the cascading effect. To tackle these challenges, we provide a cascading information chain (CIC) method to separate the independent observations from action-dependent observations and use CIC to train UNEX-RL effectively. We also discuss practical variance reduction techniques for UNEX-RL. Finally, we show the effectiveness of UNEX-RL on both public datasets and an online recommender system with over 100 million users. Specifically, UNEX-RL reveals a 0.558% increase in users' usage time compared with single-agent RL algorithms in online A/B experiments, highlighting the effectiveness of UNEX-RL in industrial recommender systems.