<p>This paper presents a novel hybrid numerical scheme for solving the multi-dimensional time-fractional convection-diffusion equations by coupling the local Floater-Hormann interpolation (LFHI) and finite difference (FD) methods. Notably, the Floater-Hormann interpolation has the unique advantage of relying only on the node distribution rather than the number of nodes, making it a meshless method. Taking advantage of this feature, we reconstruct the global Floater-Hormann interpolation differential matrix as a sparse matrix that depends only on the neighboring nodes, and achieve an efficient multi-dimensional expansion via the Kronecker product, which greatly improves the computational efficiency. We theoretically establish the error bounds of the method and rigorously prove its convergence and reliability. In our numerical validation, we provide five examples ranging from 1D to 3D to demonstrate the reliable performance of the method across different dimensions. The results show a favorable balance between accuracy and computational cost throughout these cases.</p>