"Rights" aren't a moral or ethical category. They are a juridical category.
To "mourn", on the other hand, is psychological, not moral or juridical, phenomenon.
So, the "moral right to mourn" is a conflation of disparate concepts, which becomes meaningless at all three - juridical, moral, psychological - levels.
Juridically, there is nothing that can be done to prevent people from mourning. If they think that the defeat of their preferred soccer team is something to mourn, more than the death of thousands of people in a war on the other side of the planet, there is nothing to be done about. What would we do? Jail them? Tell them they are going to hell? Dose them some chemical that will make them unable to mourn?
Morally, you can condemn whatever you want. It doesn't affect the rights of others. You may think that those who see too much television lose their moral grounds to complain about the moral decadence of society, for television is hugely responsible for such decadence. They will still watch TV and complain about the decaying mores of the commonwealth. Both things are, and should be, legal rights; a society in which either or both were forbidden would be horrible to live in.
Psychologically, there is nothing that can be done about mourning or not mourning. One may think that I should mourn the extinction of the pox virus; but the fact is that if I am not, for any reason, psychologically attached to such virus, I won't mourn its extinction, and may be indifferent or happy about it. It's possible, I guess, to shame people into pretending that they are unhappy about a given event, but it is not possible to make them unhappy if they are not.
And this - shaming people - is what this idea is probably about. It is not that we should not mourn the passing of our grandmother just because we just ate a barbecue; it is that we should not have eaten the barbecue, for we should think of the poor cow as we think of our grandmother. But as some of the comments above pointed out, the vast majority of human beings do not think a cow is equivalent to a human being, and consequently won't be able to act as if it was.
Evidently, the idea of an all-encompassing equivalence among all living beings is pragmatically unsustainable. All human beings, vegetarians and animal-rights activists included, are "especiesist" - they do not think killing a vegetal is the same as killing an animal, they do not think killing an insect - or any invertebrate - is the same as killing a mammal, they rarely think that amphibians, fish, or lizards, are on the same standing as birds and mammals, and more often than not discriminate among mammals - who does empathise with a bat or a hyaena as much as with a panda, for instance?
Talking about empathy, it is often repeated that a psychopath is someone who is devoid of empathy. But empathy is a complicated thing; as someone else put it, if you break your leg, do you want a physician who painfully pulls it into the right position, or a physician who hugs you and cries together? "Empathy" can be paralising in this sence. If no empathy makes one a psychopath, unqualified empathy may turn one hysteric.
Most of us are by far more empathetic towards our own relatives, friends, neighbours, etc, than towards people we do not know and live far away. And the idea that you should not mourn your grandmother because you didn't properly mourn the victims of a hurricane in Texas or an earthquake in Mexico is somewhat disturbing - at some point, all-encompassing empathy veers dangerously in the direction of no empathy at all.