Historians broadly agree that slavery and its expansion into western territories was the primary cause of the American Civil War. Given that, how do we assess the evolution of the conflict?

This question is a little vague, but it seems to be referring to the ways that the Union's goals of war changed over the course of the war. The reason that slave states left the Union after the election of 1860 was that Abraham Lincoln, the Republican president-elect, opposed the expansion of slavery into the western territories and states. But Lincoln could not constitutionally end slavery, and it is unlikely he would have acted to, even if that had been possible. Lincoln felt that he could not allow the Southern states to secede, and for this reason, Confederate forces fired on Union troops at Fort Sumter, beginning the war. When the war began, Lincoln's purpose was to restore the Union. As he wrote to newspaper editor Horace Greeley at the beginning of the war:

My paramount object in this struggle is to save the Union, and is not either to save or to destroy slavery. If I could save the Union without freeing any slave I would do it, and if I could save it by freeing all the slaves I would do it; and if I could save it by freeing some and leaving others alone I would also do that.

This position says more about the political realities Lincoln faced in 1861 than his personal convictions. The border states that remained in the Union—Maryland, Kentucky, and Missouri, in particular—still had slavery, and they may have accompanied the other slave states if Lincoln had immediately moved against slavery. However, by the beginning of 1863, Lincoln was forced, both as an attempt to keep England from recognizing the Confederacy, and by the thousands of enslaved people who escaped to Union lines, to issue an emancipation proclamation, which he framed as a military measure during wartime. This proclamation famously did not free enslaved people in the border states, but it did redefine the war. It meant that Union victory would result in the destruction of slavery. So as the conflict evolved, the fate of slavery depended on its outcome, and the purpose of the war went beyond simply preserving the Union.
http://www.abrahamlincolnonline.org/lincoln/speeches/greeley.htm