  This paper investigates the theory of sum-rank metric codes for which the
individual matrix blocks may have different sizes. Various bounds on the
cardinality of a code are derived, along with their asymptotic extensions. The
duality theory of sum-rank metric codes is also explored, showing that MSRD
codes (the sum-rank analogue of MDS codes) dualize to MSRD codes only if all
matrix blocks have the same number of columns. In the latter case, duality
considerations lead to an upper bound on the number of blocks for MSRD codes.
The paper also contains various constructions of sum-rank metric codes for
variable block sizes, illustrating the possible behaviours of these objects
with respect to bounds, existence, and duality properties.
