Is there any public dataset, where the historical data for index composition and constitutients are being presented?
I have only found data for S&P 500, though I would need global market index (e.g. MSCI World, S&P Global, Dow GLobal, etc.)
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I think your best bet for (free) data is to look at the holdings in a ETF that fully-replicates its benchmark.
If you go on the site for BlackRock's iShares you'll see they post the holdings for their ETFs on a monthly-basis and even allow you to download them in excel (it's also easy to edit the date in the download hyperlink). It's not ideal, but you're not going to find easily digestible, perfect data out there for free. A larger problem is this will only give you monthly data. If you want daily, you'd have to source the daily constituent returns (or market caps) between any changes and see if you could back them in to interpolate the weightings. A way around this would be to use an equal-weight index, but I have a feeling this would probably muddle the statistical significance of any inferences you may be trying to make.
Probably your best option would be calling a nearby college library and seeing if they have a Bloomberg Terminal or FactSet machine set-up. You can download the data and email it to yourself. However, be warned that Bloomberg does limit the amount of fields each terminal can download per month, so if it's popular with students it may hit its quota after the first week or so and not allow you to export in Excel format.