Oooooo! I am SO busting to write an essay on how DNA is structured and base-pairs and recombinant RNA! Sadly, it would take far more than the forum allows one person to write and would bore the contour feathers off most of our flock.
DNA (and genes) are made up of pairs of chemicals (called 'bases') that are designed to chemically attract each other and lock together when they meet. This is what keeps the DNA molecule so stable and why species remain so true to form. Changes in the pairs structure (mutations) are most often not viable and so no offspring results. VERY occasionally, a change in the base-pair structure (usually the order in which they occur) can happen in a way that doesn't change the overall shape of the molecule but which will result in some 'strange' features being observed in the offspring. This is how albinism, melanism and any number of other odd expressions of the normal situation arise.
When the reproductive cells of male and female meet (sperm and egg), an unusual process called meiosis occurs. This involves the unwinding of the DNA molecule from each cell (sperm and egg) and its reconstitution by meeting with its opposite half. That is, half the info from the male cell recombines with half the info from the female cell making a completely new entity. (
Please note: this is a highly simplified account and anyone out there who knows the actual chemistry of what happens, please bear with me, OK?) Anyway, that's how we get offspring. Half the stuff from Mum and half the stuff from Dad.
For this to happen, you have to have a perfect mesh between the two halves of the DNA molecule. That means the base-pairs on one have to line up and lock together with those on the other. If there's a gap or too many pairs on one half, there can't be a proper match and nothing can happen. (Again, highly simplified but basically what happens). The best analogy (aside from the cogs-on-a-gear one) I can think of is two long combs that can fit together and interleave their teeth. If you take two identical combs, this is easy to achieve. If you take two different brand combs, you won't get a perfect meshing together because the teeth are different sizes or different in number or spacing. Can you picture that?
So, that's what happens when you get two different species attempting to mate.
Even if a pair of birds from different species
would mate (and that's
not a given for many reasons), their DNA can't mesh together because it's too different. No amount of breeding and breeding and mutation can change that. Each time reproduction occurs, a perfectly faithful copy of the parents' DNA is made so that the essential structure (ie. that which determines a species) is maintained. For this reason, you can't 'breed out' the essential pattern of a species' DNA. The fact that the two families we're discussing here originated on different continents leads me to think it's even more remote to think a hybrid could be produced between them.
One last point. When animals breed, certain triggers are required to stimulate attraction and courting behaviour. These can include colouration, feather patterns (eg. long tails, breeding plumage), bill shape, song (especially a specific mating song, such as bower birds employ) and stage-making (again, like bower birds). This is another reason I highly doubt two such different species would even want to breed together: the correct triggers for sexual attraction are simply not there.
I dunno if this helps or not, but it's the reason I'm still puttin' all my feathers on the mutation idea.