Graph motifs are subgraph patterns that occur in complex networks, which are of key importance for gaining deep insights into the structure and functionality of the graph. Motif clustering aims at finding clusters consisting of dense motif patterns. It is commonly used in various application domains, ranging from social networks to collaboration networks, from market-basket analysis to neuroscience applications. More recently, local clustering techniques have been proposed for motif-aware clustering, which focuses on a small neighborhood of the input seed node instead of the entire graph. However, most of these techniques are designed for static graphs and may render sub-optimal results when applied to large time-evolving graphs. To bridge this gap, in this paper, we propose a novel framework, Local Motif Clustering on Time-Evolving Graphs (L-MEGA), which provides the evolution pattern of the local motif cluster in an effective and efficient way. The core of L-MEGA is approximately tracking the temporal evolution of the local motif cluster via novel techniques such as edge filtering, motif push operation, and incremental sweep cut. Furthermore, we theoretically analyze the efficiency and effectiveness of these techniques on time-evolving graphs. Finally, we evaluate the L-MEGA framework via extensive experiments on both synthetic and real-world temporal networks.