  We advance a construction for the covariant holographic entanglement
negativity for time dependent mixed states of disjoint intervals in $(1+1)$
dimensional conformal field theories ($CFT_{1+1}$) dual to bulk non static
$AdS_3$ geometries. Application of our proposal to such mixed states in a
$CFT_{1+1}$ dual to bulk non extremal and extremal rotating BTZ black holes
exactly reproduces the replica technique results in the large central charge
limit. We also investigate the time dependent holographic entanglement
negativity for such mixed states in a $CFT_{1+1}$ dual to a bulk Vaidya-$AdS_3$
geometry in the context of their thermalization involving bulk black hole
formation.
