What Statistical Tool To Measure The Weirdness Scenario:

Roughly 50 million voters vote for a certain Yes/No poll. Results
  start to come in and when the 10% of the votes are counted the difference
  between “no” and “yes” is roughly 3 million on behalf of “no”. The net
  difference (3 million) go almost unchanged until 80% of the votes are
  counted. At the end, the net difference is roughly 1.2 million on
  behalf of “no”.

I find this unnatural. Is there a way to measure how (if at all) unnatural it is?
(I chose “bias” as the tag, but not sure if it is appropriate.)
 A: Let's define the percentage of "no" as a function of time or of the share of voters that have voted (regardless of their vote) in their actual order. 
The difference in ( max - min ) values of this function can serve as the weirdness measure.  
To have a practically usable metric one might need to impose some limitations as to minimal acceptable steps (voting group size) to smooth the data or put this value in relation to the final result (percentage). 
In the situation where one is interested in how close the actual shape is to the extreme situation one can define the maximal theoretical weirdness of the dataset.  Per definition it's the maximal value of the weirdness measure over all permutations of the voting schedules.  
The ratio of the actually exhibited weirdness to the maximal theoretical weirdness could show whether there is some additional effect but the difference inbetween of voting preferences of the smallest voting groups. 
Vice versa the  selection of the voting groups in random order can show the minimal theoretically achievable weirdness. 
Etc.
Another thing to consider, to describe the weirdness of a 2D shape (the voting curve) one needs some ideal shape to compare to. If this shape is a straight line (null hypothesis?) then you could measure the area between two lines, for example. Again consider norming it so it itself becomes comparable between elections.
A: It is not quite clear what you mean by "weirdness", but a simple approach would be to define this in terms of a null-hypothesis that the votes are an unbiased sample of the population (e.g. survey-sampling).
Under the null model, the number of "yes" votes $k$ will follow a binomial distribution
$$k\sim\mathrm{Bin}(p,n)$$
where $n$ is the total number of votes tallied so far, and $p$ is the final fraction of "yes" votes (after all votes are counted).
For simplicity you can use the normal approximation
$$k\sim\mathrm{N}_{\mu,\sigma^2}\,,\,\mu=np\,\,,\sigma^2=np(1-p)$$
For a given $k$ and $n$, the margin of "no" over "yes" is
$$\hat{k}=n-2k$$
so the $z$-score of $k$ gives a simple measure of "weirdness"
$$z=\frac{k-\mu}{\sigma}$$
(note that $\hat{k}$ has the same $z$-score as $k$)
So putting it all together, a simple "weirdness" would be
$$z_{\mathrm{weird}}=\left|\frac{k-np}{\sqrt{np(1-p)}}\right|\,,\,p=\frac{K}{N}$$
where $K$ and $N$ are the population parameters (i.e. all votes tallied).
So for your example we have $p=\frac{24.4}{50}=48.8\%$, and the comparisons are (#'s in millions) ...


*

*Early: $n=0.1N=5$, $\hat{k}=3$ so $k=1$, $\mu=2.44$, $\sigma\approx{1.12}\times{10}^{-3}$ giving $|z|\approx{1288}$

*Late: $n=0.8N=40$, $\hat{k}=3$ so $k=18.5$, $\mu=19.52$, $\sigma\approx{3.16}\times{10}^{-3}$ giving $|z|\approx{323}$
So by this measure, the early results are much "weirder" than the later results, but both are very weird.
So neglecting the sequential aspect, the incremental vote-tally snapshots are not consistent with random samples from the final population.
