"The First World War came during the height of the Progressive Era. In what ways was it a 'Progressive' war?"
For the United States, World War I can be seen as a progressive war because of the country’s reasons for entering the war and because of the war aims as set out by President Woodrow Wilson.
The US did not enter WWI for about three years after the war started. The country did not feel it had a stake in the war. When it did enter, it entered in part to enforce international law. The major cause of US entry into the war was unlimited submarine warfare on the part of Germany. This was contrary to international law. The US wanted to protect the freedom of the seas. This is part of why it went to war. The desire to uphold the law and to make sure that laws are followed is a progressive idea.
Progressives also believed in the need to increase levels of democracy and in the need for fair play. These motives can be seen in the war aims of the country, particularly as seen in Wilson’s Fourteen Points. Wilson wanted to do things like ensuring that all people (or at least all Europeans) would be able to rule themselves rather than being ruled by outsiders. He wanted to have a peace in which no land would be taken from the losers of the war. He wanted a League of Nations that would protect all countries. In short, he wanted to reform the world to make for more democracy and fair treatment for all peoples.
For these reasons, WWI can be seen as a progressive war from the US perspective.