  In evaluating neoclassical transport by radially-local simulations, the
magnetic drift tangential to a flux surface is usually ignored in order to keep
the phase-space volume conservation. In this paper, effect of the tangential
magnetic drift on the local neoclassical transport are investigated. To retain
the effect of the tangential magnetic drift in the local treatment of
neoclassical transport, a new local formulation for the drift kinetic
simulation is developed. The compressibility of the phase-space volume caused
by the tangential magnetic drift is regarded as a source term for the drift
kinetic equation, which is solved by using a two-weight $\delta f$ Monte Carlo
method for non-Hamiltonian system [G.~Hu and J.~A.~Krommes, Phys. Plasmas $\rm
\textbf{1}$, 863 (1994)]. It is demonstrated that the effect of the drift is
negligible for the neoclassical transport in tokamaks. In non-axisymmetric
systems, however, the tangential magnetic drift substantially changes the
dependence of the neoclassical transport on the radial electric field $E_{\rm
r}$. The peaked behavior of the neoclassical radial fluxes around $E_{\rm r} =
0$ observed in conventional local neoclassical transport simulations is removed
by taking the tangential magnetic drift into account.
