For the S&P indices (and probably all other indices, but not 100% sure), the changes are made at the close. There are trading strategies people implement around this. My experience with this is some years back, so things may have changed.
Imagine your are the Vanguard S&P500 index tracker fund on the week that FB is getting added to the S&P 500. I don't recall the dollar value of the free floating stock for FB on the day of the add (December 20, 2013), but let's say that it was \$100B. Weights in the index are based on free floating shares - so Zuck's shares don't count. Given \$100B of free floating stock - the index tracking funds like Vanguard have about 12% of the SPX. This means that these funds need to hold 12% of the free floating shares of FB on the close on Dec 20, 2013.
In reality, they start buying some shares a few days in advance. In fact they are strongly motivated - especially on the last day - to buy shares in the open market so that they have about 50-60% of their required position before the close. Then - on the close they jam the order book with a huge market on close order which can sometimes cause the closing print to be 1-3% (I have even seen 10%) higher than the last print before the close. This is a big win for Vanguard because they are only required to replicate the closing prices - so they have, in fact, most likely gotten an average fill below the closing print with their buying and subsequent jamming on the close.
So what matters most here is the \$ amount that the indexers need to buy because their motivation for this strategy is based on the \$ amount they can make! A second order concern is what is the expected difference in average daily trading volume - this matters too, but not to the same extent.
So, buy siders sometimes (at least some years back) used to like to piggy back on these trades that the indexers were doing. Typically, such a trade would be hedged with SPX and/or the comparable stock of another company in the same industry so that macro risk is mitigated. When we were doing it, this was an amazing strategy - I couldn't understand why it just kept working, but it did. I have to imagine market efficiency has caught up by now - or rules may have changed to make it harder.
Other things to note - getting added to the S&P500 is much less interesting if the add was an upgrade from the midcap S&P 400 - in that case midcap indexers need to sell their shares - I think they used to hold 7% of the shares, so the net amount to buy was only 5% instead of 12%. This could sometimes set up good shorting opportunities if the stock moved up too much on the announcement of the add.
The key to this strategy was execution and hedging it right.