"Over time -- lots of time -- these tiny changes in the gene pool will accumulate, and after a great number of generations the population may be quite different from the ancestral stock."
Ok. I understand now.
I was assuming a change based on too narrow a scale of time.
You explanation makes a lot of sense to me.
Note that "same species as" isn't a transitive relationship. I'l use the (middle) character (of) "=" as "can interbreed with" just to shorten thing; also elipses "...". There may be a chain (geographical one have been observed) of individuals such that along the chain, neighbors can interbreed:
A = B = C = D = E
However, it does not follow that A = E. Were C to be cut out, then there would be two differing "species."
This can happen over time, A1 = A2,....A10000000=A10000001 but A1 cannot interbreed with A10000000. So even though each parent-offspring population is "of the same species," it may happen that the billionth generation would be "a different species" from the first.