The easiest and most simplistic way to look at it is to assume that the amount of genetic material halves each generation. On average, humans are about 0.1% to 0.15% different from each other, so in order to get that you'd need around $1/2^{10}=0.0976%$ and $1/2^9=0.1953%$. The exponent numbers $2^9$ and $2^{10}$ are 512 and 1024, respectively.
Armatus' Dawkins bit is almost word-for-word what he actually said:
By the time we get to third cousins, "we are getting down near the baseline probability that a particular gene possessed by A will be shared by any random individual taken from the population".
Dawkins was ball-parking that a bit, although he was really talking only about a specific gene. The inbreeding coefficient is properly defined as:
The probability that an individual carries two identical-by-descent alleles at a locus.
Which takes care of some of the fudging. With that in mind, though, within eight generations the odds are less than even you even share any genetic material with an ancestor/offspring. That's actually a mathematical analysis (with plenty of assumptions!) from the mid-80s but it's an interesting read: you can look up values for certain relations going back a bit. The answer will vary depending on the background "average" you care about. Europeans, for example, are particularly closely related.
Indeed, "[g]enetic variation is geographically structured, as expected from the partial isolation of human populations during much of their history." If you want a perhaps easier-to-read explanation of some of the math, I would suggest this 2006 review, in particular box 2. From that, a silly back-of-the-envelope calculation from 0.1% related gives a path of around 10 individuals, which is a little more distant than your third cousin.
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