Inductive knowledge graph embedding (KGE) aims to embed unseen entities in emerging knowledge graphs (KGs). The major recent studies of inductive KGE embed unseen entities by aggregating information from their neighboring entities and relations with graph neural networks (GNNs). However, these methods rely on the existing neighbors of unseen entities and suffer from two common problems: data sparsity and feature smoothing. Firstly, the data sparsity problem means unseen entities usually emerge with few triplets containing insufficient information. Secondly, the effectiveness of the features extracted from original KGs will degrade when repeatedly propagating these features to represent unseen entities in emerging KGs, which is termed feature smoothing problem. To tackle the two problems, we propose a novel model entitled Meta-Learning Based Memory Graph Convolutional Network (MMGCN) consisting of three different components: 1) the two-layer information transforming module (TITM) developed to effectively transform information from original KGs to emerging KGs; 2) the hyper-relation feature initializing module (HFIM) proposed to extract type-level features shared between KGs and obtain a coarse-grained representation for each entity with these features; and 3) the meta-learning training module (MTM) designed to simulate the few-shot emerging KGs and train the model in a meta-learning framework. The extensive experiments conducted on the few-shot link prediction task for emerging KGs demonstrate the superiority of our proposed model MMGCN compared with state-of-the-art methods.