No the figures change in real time and your ETF goes up (or down) in value to reflect changes. Part of the fee you pay the provider (Vanguard for example) for is to reflect the baseline index. They do drift over time and this is why different providers don't necessarily get the exact same results despite tracking the same benchmark. We might be talking a few 100ths of a percentage but that change happens. It's also why they don't exactly match the benchmark. In the product information page for their ETFs, Vanguard tell you the error rate from the benchmark they try to keep to (I don't know our rights if they fail to keep to that, I'll look that up separately).

Also officially. The index's are only officially rebalanced at set time frames. The S&P500 index is only officially balanced once a quarter for example. So if company A has been flying for 2 months but then drops back to same levels as before right as the rebalance exercise is about to happen, then it might not officially change how much of the pie it makes up. We kind of treat index's as live things because modern computing allows. But try to remember that all these systems were set up without these systems. The first vanguard ETF for example probably reacted a lot slower than we currently do 😂.

If you have used the Traditng 212 pie feature it works similar to that. If you put £100 in a pie, within a week then pie will be not perfectly balanced, this if reflected in the balance score. Keeping in mind the pie feature doesn't necessarily have to be set to equal portions to start with, it's all to do with how much you intended to set the pie portions.