Thus, said change is passed along to subsequent generations until the change(s) are enough to be considered a new species that branched off of the parent one.
If there is no granted advantage...the trait usually just dies out.
There has been no need to evolve. None of the random genetic changes we have seen have been advantageous to the species. In fact...they're usually harmful (food allergies, autoimmune disorders, etc).
We as humans have not managed to come across anything that we needed to evolve a coping mechanism for. If it gets cold, animals evolve more fur. We just turn up the heat. If it gets hot, animals lose the fur, we just turn up the AC. Too bright? We invent sunglasses.
There is no reason for us to evolve and so we will not. It's rather like how sharks have remained unchanged since long before mankind evolved.
In your face Creationists.
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Randy the Atheist say: A species evolves only when something begins to pulverize it - be it the weather or a new invasive predator.
In a nutshell, evolution is caused by a malfunction in the DNA processes due to its error-prone replication system which causes small or large mutations in the new copies of itself. These mutated copies are then thrown at Natural Selection where it tries to exterminate them into oblivion by a number of tools like miscarriage, starvation, sickness or sexual avoidance.
Successful evolution occurs when a species generates an accidental mutation that just so happens to evade the Natural Selection Blender of Death. It doesn't have to be the fittest. It just has to have the right mutation at the right moment.
Most of these evolutionary mutations come about when a species is being pulverized by something - be it the weather or a new invasive predator. The smaller populations make inbreeding corruption rates significantly higher which makes mutation rates skyrocket. This high rate of mutation causes better chances for evolutionary adaptation at the expense of death, debilitation and disease which Natural Selection happily destroys into oblivion. These failed lines of genetic experimentation are what we see on the charts as extinction. The lines that continue are what we see as transitionals - random genetic accidents that shuffled the right amount of DNA into the right places.