How do changes in oil prices impact the global economy?

Even though oil is a less important production input than it was three decades ago, that reasoning should work in reverse when oil prices fall, leading to lower production costs, more hiring, and reduced inflation. But this channel causes a problem when central banks cannot lower interest rates.
Crude oil is a major economic input, so a rise in oil prices contributes to inflation, which measures the overall rate of price increases across the economy.