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Large-scale cross-lingual pre-trained language models (xPLMs) have shown effectiveness in cross-lingual sequence labeling tasks (xSL), such as cross-lingual machine reading comprehension (xMRC) by transferring knowledge from a high-resource language to low-resource languages. Despite the great success, we draw an empirical observation that there is a training objective gap between pre-training and fine-tuning stages: e.g., mask language modeling objective requires local understanding of the masked token and the span-extraction objective requires global understanding and reasoning of the input passage/paragraph and question, leading to the discrepancy between pre-training and xMRC. In this paper, we first design a pre-training task tailored for xSL named Cross-lingual Language Informative Span Masking (CLISM) to eliminate the objective gap in a self-supervised manner. Second, we present ContrAstive-Consistency Regularization (CACR), which utilizes contrastive learning to encourage the consistency between representations of input parallel sequences via unsupervised cross-lingual instance-wise training signals during pre-training. By these means, our methods not only bridge the gap between pretrain-finetune, but also enhance PLMs to better capture the alignment between different languages. Extensive experiments prove that our method achieves clearly superior results on multiple xSL benchmarks with limited pre-training data. Our methods also surpass the previous state-of-the-art methods by a large margin in few-shot data settings, where only a few hundred training examples are available.