  We advance a covariant holographic construction based on the minimal
entanglement wedge cross section, for the entanglement negativity of bipartite
states in $CFT_{1+1}$s dual to non static bulk $AdS_3$ geometries. In this
context we obtain the holographic entanglement negativity for bipartite states
in $CFT_{1+1}$s dual to bulk non extremal and extremal rotating BTZ black holes
and the time dependent Vaidya-$AdS_3$ geometries. Our results correctly
reproduce the replica technique results for the corresponding dual $CFT_{1+1}$s
in the large central charge limit. Furthermore they also match exactly with the
corresponding results from an alternate earlier covariant holographic
entanglement negativity proposal based on an algebraic sum of the lengths of
bulk extremal curves (geodesics) homologous to specific combinations of
intervals for the configurations in question. This demonstrates the equivalence
of the two constructions and indicates that the algebraic sum of the lengths of
such bulk extremal curves (geodesics) is proportional to the minimum
entanglement wedge cross section for the $AdS_3/CFT_2$ scenario, which possibly
extends to higher dimensions.
