In this paper, we study Lagrangian mean curvature flow (LMCF) of asymptotically conical (AC) Lagrangian submanifolds asymptotic to a union of special Lagrangian cones. Since these submanifolds are non-compact, we establish a short-time existence theorem for AC LMCF first. Then we focus on the equivariant, almost-calibrated case and prove long-time existence and convergence results. In particular, under certain smallness assumptions on the initial data, we show that the equivariant, almost-calibrated AC LMCF converges to a Lagrangian catenoid or an Anciaux’s expander.