<p>We construct a pullback measure attractor for the stochastic nonautonomous nonlocal parabolic equation with nonlinear multiplicative noise, in the proof, we use a two-stage approach which is different from recent publications for other models, and we also use a special technique to estimate the moment errors for the nonlocal equations. We then show that the pullback measure attractor is backward tempered, backward tight and backward bounded in the measure phase spaces. Moreover, we prove the upper semi-convergence of pullback measure attractors towards the measure attractor when the stochastic process of external forces converges to the time-independent random force, and we also show the convergence from evolutionary measures to the invariant measure when the time tends to the minus infinity. It seems to be the first time to establish such asymptotic autonomy for evolution systems of probability measures.</p>