Despite the tremendous progress recently made towards automatic sleep staging in adults, it is currently known if the most advanced algorithms generalize to the pediatric population, which displays distinctive characteristics in overnight polysomnography (PSG). To answer the question, in this work, we conduct a large-scale comparative study on the state-of-the-art deep learning methods for pediatric automatic sleep staging. A selection of six different deep neural networks with diverging features are adopted to evaluate a sample of more than 1,200 children across a wide spectrum of obstructive sleep apnea (OSA) severity. Our experimental results show that the performance of automated pediatric sleep staging when evaluated on new subjects is equivalent to the expert-level one reported on adults, reaching an overall accuracy of 87.0%, a Cohen's kappa of 0.829, and a macro F1-score of 83.5% in case of single-channel EEG. The performance is further improved when dual-channel EEG$\cdot$EOG are used, reaching an accuracy of 88.2%, a Cohen's kappa of 0.844, and a macro F1-score of 85.1%. The results also show that the studied algorithms are robust to concept drift when the training and test data were recorded 7-months apart. Detailed analyses further demonstrate "almost perfect" agreement between the automatic scorers to one another and their similar behavioral patterns on the staging errors.