A large majority of approaches have been proposed to leverage the dependency tree in the relation classification task. Recent works have focused on pruning irrelevant information from the dependency tree. The state-of-the-art Attention Guided Graph Convolutional Networks (AGGCNs) transforms the dependency tree into a weighted-graph to distinguish the relevance of nodes and edges for relation classification. However, in their approach, the graph is fully connected, which destroys the structure information of the original dependency tree. How to effectively make use of relevant information while ignoring irrelevant information from the dependency trees remains a challenge in the relation classification task. In this work, we learn to transform the dependency tree into a weighted graph by considering the syntax dependencies of the connected nodes and persisting the structure of the original dependency tree. We refer to this graph as a syntax-transport graph. We further propose a learnable syntax-transport attention graph convolutional network (LST-AGCN) which operates on the syntax-transport graph directly to distill the final representation which is sufficient for classification. Experiments on Semeval-2010 Task 8 and Tacred show our approach outperforms previous methods.