The short answer is that all three indices are designed to be broad measures of how U.S. stocks are performing. So, if they're meeting (or even coming close to fulfilling) that objective, then you would expect to see the same patterns (but not necessarily the same magnitudes) in each of them.
There are differences in how each index attempts to fulfill the objective of tracking U.S. markets, though. For example, the DJIA tracks only 30 prominent companies. Thus, news that is specific to one of those companies will generally affect the Dow more than the S&P 500 which, as the name suggests, tracks 500 different companies. Of course, a stock comprising 1/30 of an index will have much more impact on the index than one that comprises only 1/500 of an index.
However, news that impacts entire large industries or, especially, the U.S. economy as a whole will generally affect all three indices in similar ways, since a broad section of the component stocks of all three will be affected.