Gradient-based meta-learning (GBML) with deep neural nets (DNNs) has become a popular approach for few-shot learning. However, due to the non-convexity of DNNs and the bi-level optimization in GBML, the theoretical properties of GBML with DNNs remain largely unknown. In this paper, we first aim to answer the following question: Does GBML with DNNs have global convergence guarantees? We provide a positive answer to this question by proving that GBML with over-parameterized DNNs is guaranteed to converge to global optima at a linear rate. The second question we aim to address is: How does GBML achieve fast adaption to new tasks with prior experience on past tasks? To answer it, we theoretically show that GBML is equivalent to a functional gradient descent operation that explicitly propagates experience from the past tasks to new ones, and then we prove a generalization error bound of GBML with over-parameterized DNNs.