This paper presents a method for the joint detection and tracking of weak targets in automotive radars using the multi-frame track-before-detect (MF-TBD) procedure. Generally, target tracking in automotive radars is challenging due to radar field of view (FOV) misalignment, nonlinear coordinate conversion, and self-positioning errors of the ego-vehicle, which are caused by platform motion. These issues significantly hinder the implementation of MF-TBD in automotive radars. To address these challenges, a new MF-TBD detection architecture is first proposed. It can adaptively adjust the detection threshold value based on the existence of moving targets within the radar FOV. Since the implementation of MF-TBD necessitates the inclusion of position, velocity, and yaw angle information of the ego-vehicle, each with varying degrees of measurement error, we further propose a multi-frame energy integration strategy for moving-platform radar and accurately derive the target energy integration path functions. The self-positioning errors of the ego-vehicle, which are usually not considered in some previous target tracking approaches, are well addressed. Numerical simulations and experimental results with real radar data demonstrate large detection and tracking gains over standard automotive radar processing in weak target environments.