Even if all that is true. I think it's quite a leap to characterize their behavior as "mourning" as in the same a people. I think you're attributing human traits to them that they aren't able to relate to.
Researchers believe that much of our emotional makeup is chemical. Many animals share the same chemicals as humans.
A mother, after birth and during suckling, produces oxytocin and prolactin and these chemicals help her bond to her new baby. Her hormones help to produce a loving feeling towards her infant. Other mammals produce these same hormones. If chemical reactions help produce a loving feeling within a human, could they not also produce the same feelings in other mammals?
There is all sorts of research being done on the emotional lives of animals. All you have to do is google.
http://www.jeffreymasson.com/animal-books/when-elephants-weep.html