  We compare finite rank perturbations of the following three ensembles of
complex rectangular random matrices: First, a generalised Wishart ensemble with
one random and two fixed correlation matrices introduced by Borodin and
P\'ech\'e, second, the product of two independent random matrices where one has
correlated entries, and third, the case when the two random matrices become
also coupled through a fixed matrix. The singular value statistics of all three
ensembles is shown to be determinantal and we derive double contour integral
representations for their respective kernels. Three different kernels are found
in the limit of infinite matrix dimension at the origin of the spectrum. They
depend on finite rank perturbations of the correlation and coupling matrices
and are shown to be integrable. The first kernel (I) is found for two
independent matrices from the second, and two weakly coupled matrices from the
third ensemble. It generalises the Meijer $G$-kernel for two independent and
uncorrelated matrices. The third kernel (III) is obtained for the generalised
Wishart ensemble and for two strongly coupled matrices. It further generalises
the perturbed Bessel kernel of Desrosiers and Forrester. Finally, kernel (II),
found for the ensemble of two coupled matrices, provides an interpolation
between the kernels (I) and (III), generalising previous findings of part of
the authors.
