France wanted payback for the Franco-Prussian War (which Napoleon III started), including the return of Alscace-Lorraine.
Britain was looking for an excuse to take out the German Navy, which was getting too big for Britain's liking. Britain was determined to maintain global naval supremacy.
The Tsar didn't want war. But Russia was seen as the great protector of Slavs. The Tsar felt he had to defend Serbia to maintain face as a world power, especially after Russia's defeat in the Russo-Japanese War.
But Russia was seen as the great protector of Slavs.
Ask the Poles as to how Russia “protected Slavs”. There’s a reason that the Polish National Anthem mentions Napoleon in it.