Does the price of an index (e.g. Dow Jones) go down on an ex-dividend date?

Surprisingly web search returns results for stock prices (they obviously go down other things equal in case of large dividend).

But I've always assumed indexes are adjusted for dividend payouts (so index opens same value on ex-dividend date, not dips down). Is it correct for all major indexes or not?

Was it the same way before, e.g. near time of the Great Depression, 1930s?