In this work, the dynamics of a prototypical heavy-light-heavy abstract reaction, Cl(2P) + HCl → HCl + Cl(2P), is investigated both by constructing a new potential energy surface (PES) and by rate coefficient calculations. Both the permutation invariant polynomial neural network method and the embedded atom neural network (EANN) method, based on abinitio MRCI-F12+Q/AVTZ level points, are used for obtaining globally accurate full-dimensional ground state PES, with the corresponding total root mean square error being only 0.043 and 0.056 kcal/mol, respectively. In addition, this is also the first application of the EANN in a gas-phase bimolecular reaction. The saddle point of this reaction system is confirmed to be nonlinear. In comparison with both the energetics and rate coefficients obtained on both PESs, we find that the EANN is reliable in dynamic calculations. A full-dimensional approximate quantum mechanical method, ring-polymer molecular dynamics with a Cayley propagator, is employed to obtain the thermal rate coefficients and kinetic isotopic effects of the title reaction Cl(2P) + XCl→ XCl + Cl(2P) (H, D, Mu) on both new PESs, and the kinetic isotope effect (KIE) is also obtained. The rate coefficients reproduce the experimental results at high temperatures perfectly but with moderate accuracy at lower temperatures, but the KIE is with high accuracy. The similar kinetic behavior is supported by quantum dynamics using wave packet calculations as well.