Strongly peaked tungsten accumulation is a common feature of high performance plasma scenarios in JET with the ITER-like wall, particularly during MHD activity induced by continuous modes. This study investigates the effect of long living internal kink modes on heavy impurity transport in the presence of strong flows and NTV ambipolar electric field. A novel formulation which includes these effects is presented and applied in the VENUS-LEVIS code in order to follow tungsten ions in a saturated JET-like internal kinked toroidally rotating plasma configuration. The synergy between 3D magnetic fields, strong flows and NTV is seen to cause tungsten accumulation in contrast to what is observed in similar axisymmetric configurations. Rapid inward transport of impurities in JET plasmas following the triggering of continuous modes is explained by the work presented here, and we use the same theory to postulate why outward transport can occur in kinked ASDEX-U plasmas.