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Nowadays, we are witnessing an increasing effort to improve the performance and trustworthiness of Deep Neural Networks (DNNs), with the aim to enable their adoption in safety critical systems such as self-driving cars. Multiple testing techniques are proposed to generate test cases that can expose inconsistencies in the behavior of DNN models. These techniques assume implicitly that the training program is bug-free and appropriately configured. However, satisfying this assumption for a novel problem requires significant engineering work to prepare the data, design the DNN, implement the training program, and tune the hyperparameters in order to produce the model for which current automated test data generators search for corner-case behaviors. All these model training steps can be error-prone. Therefore, it is crucial to detect and correct errors throughout all the engineering steps of DNN-based software systems and not only on the resulting DNN model. In this paper, we gather a catalog of training issues and based on their symptoms and their effects on the behavior of the training program, we propose practical verification routines to detect the aforementioned issues, automatically, by continuously validating that some important properties of the learning dynamics hold during the training. Then, we design, TheDeepChecker, an end-to-end property-based debugging approach for DNN training programs. We assess the effectiveness of TheDeepChecker on synthetic and real-world buggy DL programs and compare it with Amazon SageMaker Debugger (SMD). Results show that TheDeepChecker's on-execution validation of DNN-based program's properties succeeds in revealing several coding bugs and system misconfigurations, early on and at a low cost. Moreover, TheDeepChecker outperforms the SMD's offline rules verification on training logs in terms of detection accuracy and DL bugs coverage.