Diffusion models have recently been adopted for point cloud upsampling due to their effectiveness in solving ill-posed problems. However, existing upsampling methods often struggle with inefficiencies, as they generate dense point clouds by mapping Gaussian noise to data, overlooking the geometric information already present in sparse inputs. To address this, we propose PUFM, a novel Point Cloud Upsampling via Flow Matching, which learns to directly transform sparse point clouds into their high-fidelity dense counterparts. Our approach first applies midpoint interpolation to densify the sparse input. Then, we construct a continuous interpolant between sparse and dense point clouds and train a neural network to estimate the velocity field for flow matching. Given the unordered nature of point clouds, we introduce a pre-alignment step based on Earth Mover's Distance (EMD) optimization to ensure coherent and meaningful interpolation between sparse and dense representations. This results in a more stable and efficient learning trajectory during flow matching. Experiments on synthetic benchmarks demonstrate that our method delivers superior upsampling quality but with fewer sampling steps. Further experiments on ScanNet and KITTI also show that our approach generalizes well to real-world RGB-D and LiDAR point clouds, making it more practical for real-world applications.