The Standard Model has point like leptons and quark. The quarks interact via the strong interaction, which is QCD. QCD conserves flavor. That is, up quarks stay up quarks, same for down, strange, charm, bottom, and top flavors.
So now suppose the quarks are composite. That means QCD is no longer the fundamental theory. You have to suppose there is a new gauge theory, with a very short range. Hence, the term "contact-interaction". This is how the weak interaction was handled at low energy long before their were gauge bosons to exchange: it's just a 4-point interaction term.
On to flavor-diagonal: First lets back-up and look at this whole processes of discovering something is composite. We have been through it with the neutron and proton. Originally,
neutrons and protons were seen as fundamental particles interacting via a pion field. As thing progressed, it was released the neutron and proton were made of quarks, and the pion was a quark/antiquark pair. It turns out, neutrons and protons could transmute into each other:
Note the following (virtual) reaction:
$$ n \rightarrow p + \pi^- $$
That is an isospin changing reaction (for the baryon part), even though the underlying theory conserved flavor (whence isospin emerges).
Now back to out composite quarks: we don't have flavor changing reactions at the quark level, so we need to imagine a composite theory that doesn't change flavor, even with composite interactions.
At this point you say flavor-off-diagonal processes are suppressed (even if you don't have an explanation), and look only at flavor diagonal processes. These are the processes that move quark constituents around without changing the flavor of the quarks.
The good news is, LHC turned on, and quarks are still point-like.