President Obama has sought to package the American airstrikes against Islamic State militants in Iraq and Syria as a single straightforward battle, to “degrade and destroy” an organization that has accelerated the destabilization of the region and poses a potential, though not imminent, threat to the United States. On Wednesday, American military officials again reported attacks on Islamic State targets, describing strikes against oil refineries controlled by the group in eastern Syria.

But unleashing American firepower against the Islamic State introduces new, unpredictable dynamics that vary from place to place in the volatile and increasingly fragmented conflict. Hitting the Islamic State means different things to different players in different places — adding to the difficulties Mr. Obama faces in building a coherent coalition against the group.

Syrian insurgents, including some who the Obama administration says will act as its ground force against the Islamic State, have criticized the strikes, while one official in the Syrian government, which has condemned the strikes as a violation of its sovereignty, told Reuters they seemed to be “proceeding in the right direction.”

The fighting in Kobani, a collection of Kurdish villages whose cinder-block buildings can be seen just beyond the border posts, is just one of the many wars within the war that have tangled alliances across the region.

Kurds in the area, home to about 400,000 people, had largely ruled themselves for the past two years after government forces withdrew, allowing a Kurdish party to take over military and government posts. The party’s military wing is affiliated with the separatist Kurdistan Workers’ Party in Turkey; the PKK, as the party is also known, is a bitter enemy of the Turkish government, which considers it a terrorist organization.