  We consider $2$-designs which admit a group of automorphisms that is
flag-transitive and leaves invariant a chain of nontrivial point-partitions. We
build on our recent work on $2$-designs which are block-transitive but not
necessarily flag-transitive. In particular we use the concept of the ``array''
of a point subset with respect to the chain of point-partitions; the array
describes the distribution of the points in the subset among the classes of
each partition. We obtain necessary and sufficient conditions on the array in
order for the subset to be a block of such a design. By explicit construction
we show that for any $s \geq 2$, there are infinitely many $2$-designs
admitting a flag-transitive group that preserves an invariant chain of
point-partitions of length $s$. Moreover an exhaustive computer search, using
{\sc Magma}, seeking designs with $e_1e_2e_3$ points (where each $e_i\leq 50$)
and a partition chain of length $s=3$, produced $57$ such flag-transitive
designs, among which only three designs arise from our construction -- so there
is still much to learn.
