  The aim of this research is to introduce the notion of multi-transitivity in
non-autonomous discrete dynamical systems(NDDS) with respect to a vector.
Necessary and sufficient conditions are obtained under which a NDDS is
multitransitive and strongly multi-transitive. Examples and counter-examples
are constructed to justify that results in Remark 3.7 [1] what is true in
autonomous discrete dynamical systems(ADDS) need not be true in NDDS. Also, an
affirmative answer to the question posed by Salman and Das [9] is given.
Finally, the Furstenberg family generated by a vector characterises
multi-transitivity with respect to a vector for NDDS is obtained.
