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This is off the top of my head. If I am in error, it’s just me being ignorant.

Evolution is sometimes defined as change in allele frequency in populations. This can be due to selection, or it can be due to genetic drift.

There are other, related processes involved. There are a dozen or so ways that genes can be altered. They can also be duplicated or be inserted by viral infection. Genes can migrate form one chromosome to another. Chromosomes can also undergo duplication and various kinds of changes.

Evolutionary change always involves a change to the genome in a reproducing individual. The change must be carried forward to offspring. Changes to genes may or may not produce somatic changes, but if they have a statistical influence on the probability of surviving and reproducing, they will be subject to selection. If their effect is neutral, they can still diffuse through a population.

Traits that are visible and present in nearly all individuals in a population, over thousands of generations, are presumed to be stabilized by selection. Individuals lacking the trait are statistically less likely to survive and produce offspring. Selection is not simply a matter of which individuals are stronger. Female choice is a well documented mode of selection. Selective breeding by humans is another. Both of these produce change in a population faster than other known kinds of selection.

To summarize: evolution is an iterative process that involves genetic change in individuals, and differential reproductive success.

That’s pretty good, but it’s an answer focused only small scale processes - that’s only part of the story, so let me clarify a few items. I’m sorry for the long post, but I hope you and others find it helpful.

Evolution is not really just about “change over time”...I have changed over time but that’s not evolution. That’s just aging :-p. Lots of things “change over time”. So, a more precise definition of “evolution” is that it is descent with modification.

This definition allows for the small scale, or “micro-scale”, processes you described as well as the larger or macro-scale processes that deal with the descent of various organisms from a common ancestor. It is the assertion of evolutionary theory that it is the small scale process — the thing that happens at the genetic level — that eventually manifests in changes at the larger scale over time.

OK. So, what causes changes at the small scale? You said “natural selection” or genetic drift. That’s good, that’s part of it, but that’s not all there is.
As you said, evolution is supposed to happen when you have a change in the frequency of an allele in a population — for example, more and more Blue-footed Hairless Dingnuts being born with an allele for hair than the allele that leaves them without hair. Eventually, so the story goes, there will be populations of hairless dingnuts and separate populations of dingnuts with hair. More time will pass, accompanied by more changes in the expression of other alleles, and there will be hairless dingnuts, dingnuts with hair, and dingnuts with hair and red feet and long tails. This latter group will be unable to successfully reproduce with the first two groups — leading to a new species — and we’ll eventually end up with all sorts of terrestrial dingnuts, aquatic dingnuts, perhaps some dingnuts that fly, and new groups of things that aren’t even dingnuts at all anymore.

The idea is that millions of very different organisms were all thusly able to descend from a common ancestor.

Anyway. So, how do these changes happen in the first place? There are 4 posited mechanisms through which such changes could occur:

1. Mutation
2. Migration
3. Genetic drift, which you mentioned
4. Natural selection, which you also mentioned

Of these, mutation is the most fundamental because without it none of the other mechanisms can be set into motion. Mutation is fairly self-explanatory — whoa, hey look Ethel, junior’s got an extra fin and it’s longer than all the others! It’s totally and completely dependent on chance.

Migration sometimes happens by chance, sometimes under the direction of some external event (a drought, say). Genetic drift is also mediated by chance - say that in a bunch of our original mixed group of hairless and newly hairy dingnuts there were a bunch of hairless dingnuts who died without having offspring. Statistically, this means that there are now more hairy dingnuts than there are hairless dingnuts having offspring. Genetic drift...like mutation, happens by chance.

And this — this idea of chance — is an important point. A really, really important point.

Every bit of evolutionary change that is supposed to have ever occurred has started with...an accident (mutation) and been moderated by 3 other mechanisms which are also heavily, heavily skewed toward chance. So, this means...

Somehow, chance — and, really, in its simplest analysis, only chance — must account for all of the life in all its many splendored colors and forms on the planet today.

Not likely. Not at all likely, but let me leave that aside for a moment. Let me make the huge leap of faith that it is possible for chance to accomplish that feat — it’s a bit further for me to go than the leap of faith to believe in God, but I think I can do it.

You asked me what natural processes required for evolution have yet to be observed through scientific methods. The answer is...chance. The most basic, fundamental spark that lights the flame for all the rest of the hypothetical mess that is the idea of evolution.

The very first principle of the scientific method is observation. And chance - the driver for mutation and genetic drive - simply fails the test. Chance is NOT an observable natural process like, say, erosion. No one has ever observed it in the field, and no one has ever been able to collect any data from a “mutation in action”.

Furthermore, we can’t test evolution - we can’t test chance, we can’t repeat it - not in the field, not in the lab. Some have tried to simulate chance - using random number generators and all sorts of other scientific methods to try to show that, yes, it is possible to start with species A and eventually end up at species B...but no one has succeeded.

So we can’t observe evolution in action, we can’t test it, and we can’t replicate it.

All...ALL...that we can do is observe what we THINK are its end products. And that means running a lot of analyses to try and show relationships between current species on the planet — try to construct the family tree — and collecting a lot of fossils to try and fill in the historic links from the past.

Both efforts have failed miserably. Of the millions of species that have lived on the planet you’d think we’d have gotten a more reliable set of phylogenies together by now. But the truth is that they’re all full of holes...big holes...right where you’d least expect them to be. And the fossil record is no help either. All we’ve got to show is a handful of bones - a jawbone here, a nearly intact pre-hominid there, and some DNA tests that don’t actually show what it was hoped they would. I simplify, I know, but providing more detailed examples really doesn’t improve the picture.

Anyway, I’ll end with that for the sake of the length of this post. But “evolution” simply does not offer the set of proofs that you and so many others on this board hope that it does.