Applying convolution methods to domains that lack regular underlying structures is a challenging task for 3D vision. Existing methods require the manual design of feature representations suitable for the task or full-voxel-level analysis, which is memory intensive. In this paper, we propose a novel feature extraction method to facilitate 3D nonrigid shape analysis. Our approach, called 3D-MConv, extends convolution operations from regular grids to irregular mesh sets by parametrizing a series of convolutional templates and adopts a novel local perspective to ensure that the algorithm is more invariant against global isometric deformation and articulation. We carefully design the convolutional template as a polynomial function that flexibly represents the local shape. An unsupervised learning method is adopted to learn the convolutional template function. By using the convolution operation and the movement of the template on the model surface, we can obtain the distribution of the typical template shapes. We combine this distribution feature with the spatial co-occurrence information of typical template shapes modelled by Markov chains to form a high-level descriptor of a 3D model. The support vector machine method is used to classify the nonrigid 3D objects. Experiments on SHREC10 and SHREC15 demonstrate that 3D-MConv achieves state-of-the-art accuracy on standard benchmarks.