As machine learning models are increasingly deployed in production, robust monitoring and detection of concept and covariate drift become critical. This paper addresses the gap in the widespread adoption of drift detection techniques by proposing a serverless-based approach for batch covariate drift detection in ML systems. Leveraging the open-source OSCAR framework and the open-source Frouros drift detection library, we develop a set of services that enable parallel execution of two key components: the ML inference pipeline and the batch covariate drift detection pipeline. To this end, our proposal takes advantage of the elasticity and efficiency of serverless computing for ML pipelines, including scalability, cost-effectiveness, and seamless integration with existing infrastructure. We evaluate this approach through an edge ML use case, showcasing its operation on a simulated batch covariate drift scenario. Our research highlights the importance of integrating drift detection as a fundamental requirement in developing robust and trustworthy AI systems and encourages the adoption of these techniques in ML deployment pipelines. In this way, organizations can proactively identify and mitigate the adverse effects of covariate drift while capitalizing on the benefits offered by serverless computing.