Inflation is closely monitored by central banks as well. Whenever inflation accelerates at too high a rate, a central bank can increase interest rates to cool down the economy. Increased rates decrease spending and borrowing, which further caps inflation but slows down the economic growth too. This is the biggest challenge of a central bank since they need to measure the conditions in an economy and respond accordingly. The global factors further impact the decisions of the central bank, which include changes in the policies of developed countries or international economic trends.
But another point is that central banks are part of an international system. The actions of one central bank, due to interdependence among economies, can have implications for others.