Other genomes will be selected for increased complexity and will acquire new traits over time as you say.
If we are talking about all life, again it's tough to make a prediction. In general I would think the diversity would increase, but it would depend on the time frame being looked at. In periods where life is flourishing I would expect more genes to be added over time, in periods of mass extinctions the genetic diversity might be decreasing.
Talking about comparing the earliest organisms, say the last common ancestor of plants and animals or LCA of animals with how evolution is thought to proceed under NeoDarwinism.
"The proponents of the synthetic theory maintain that all evolution is due to the accumulation of small genetic changes, guided by natural selection, and that transspecific evolution is nothing but an extrapolation and magnification of the events that take place within populations and species." (Mayr 1963)
I think the quote above expresses the general ND narrative, namely "a slow accumulation of small genetic changes, guided by natural selection." A number of years back, I ran across the concept put forward by front loaders or those like Pierre Grasse that believed NeoDarwinian methods cannot explain macroevolution. Their prediction, at least some of them, was that the LCA would be genetically more complex and that evolution does not proceed through a slow accumulation of genes.
Of course, this was a somewhat radical proposition and flew in the face of the idea that genes accrue slowly via NeoDarwinian means. However, recent studies amazingly proved the front loaders right in their predictions and standard evo wrong.
Here is a comment by the author of one of those studies. Note: he is not abandoning Neodarwinism but does recognize his findings turn over some common preconceptions by evolutionists.
However, Professor David Miller says its genetic complexity challenges the notion that life started out simple then evolved to become more sophisticated.
"There's this intrinsic tendency to think about a slow accumulation of complexity and a slow accumulation of genes which have allowed an increased morphological complexity in higher animals and what the coral genomes tell us is that that's completely wrong and that most genes were invented very early in animal evolution," he said.
http://www.jcu.edu.au/cgc/MillerHP.html
His isn't the only work showing this. Of course, evos will now likely insist either the LCA evolved all this genetic complexity, even genes for human nerve function in corals for example, slowly over time via NeoDarwinian methods while still being a very primitive organism in it's cellular organization, and then we saw a "massive loss of genes for some lineages", etc,....or that the LCA was actually very complex morphologically.
Of course, we can test neither of those claims.
What we do know is this a failure of one side, standard evolutionism, in it's predictions and a successful prediction of another, the front loaders. Both adhere to common descent, and of course an alternative is still special creation, but would like to limit the scope of this discussion a bit (to these specific predictions under the assumption of common descent) or we might just spin our wheels in too broad of a topic.
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