Do ETFs move on their own? Or only on aggregate from the individual holdings?

ETFs have shares that are traded on the market just like any other security. Their price rises or falls with demand; as such, it's not directly linked to the underlying index. The linkage is indirect, though it's quite strong of course: the ETF's assets are the shares of the securities that make up the index that it tracks, and so if it is executed correctly and with relatively little overhead, it should very nearly track the index. (There is always some overhead, in transaction fees if nothing else, so it's never perfect, but an efficient ETF gets very close.) Investopedia's explanation of ETFs is worth a read if you want more information.

Note that an ETF literally means "exchange traded fund", which means it is a fund that is bought and sold in this manner - as opposed to a "mutual fund". It doesn't even have to be an index tracking fund - they just usually are; that's a common misconception, though. There are mutual funds that also track indexes; you can read more about the difference between the two here.

X, A, B and C all trade independently and are subject to their own buy and sell pressures that affects their prices. But the price of the ETF tends to be consistent with the price of the underlying shares, for a couple of reasons.

The fundamental value of X is just the aggregated value of the underlying shares. After all it's just (indirect) ownership of shares of A B and C. Who would buy X if it's trading at a large premium compared to A, B and C?

But, more importantly, authorized participants can hand in one share of X to the ETF provider and receives shares of A, B and C in exchange. They can do the opposite too, they can hand in shares of A, B and C to receive one new share of X.

Therefore, if X trades at a discount compared to A, B and C, an authorized market maker can make a profit by buying shares of X, redeeming them for A, B and C, and selling those. Which will drive the price of X up, and the prices of A, B and C down, thus causing them to converge. In this way the sell pressure on X is "passed through" to A, B and C. The reverse happens if X trades at a premium.

This process is not instantaneous or friction-less so the price of the ETF will not necessarily match the underlying shares exactly at any given time. But significant differences tend to be arbitraged away.

I would say it's both true that the price of X affects the prices of A, B and C, and the prices of A, B and C affect the price of X. The prices will tend to converge because of arbitrage but in general it's difficult to say which caused which.

Of course the relationship is not quite symmetrical, because X derives it's fundamental value from A, B and C, the reverse is not true.

The most common arb is the creation and redemption mechanism by an authorized participant who buys the index securities and swaps them for ETF shares (creation) and vice versa for redemption.

The creation/redemption arb is beyond the means of non-institutional participants. Their alternative is to pairs trade on ETFs on the same underlying index where one trades at a premium or discount to the other.