The Nez Perce War originated from an attempt to drive the Nez Perce Native Americans onto a smaller segment of their original tribal lands in the Idaho territory. A significant portion of the Nez Perce nation refused, and fled north seeking aid first from the Crows and then from Sitting Bull in Canada.
To the best of my knowledge, the nontreaty Nez Perce intended to flee the United States altogether. I understand that there was a bout of pre-war raiding which killed several settlers, but over the course of the war the Nez Perce appear to have been in retreat and relatively passive towards settlers.
Why was so much effort expended to track down a group attempting to leave the United States and force them on to the reservation? My impression is that the Nez Perce were considered an obstacle to settlement of the West, so what was the justification for bringing them back to the United States instead of allowing them to depart to Canada?
(Or, perhaps: were Miles and Howard aware of the Nez Perce intent to leave the country?)
Hello. I'm a mod over on the most active Native American subreddit, /r/IndianCountry. I am also Nez Perce and have looked into this topic frequently. I am by no means a historian, though, and am open to criticism.
This is an interesting question and, like with many things, it requires a bit of context to fully understand the situation during this time.
Background
In 1855, Old Joseph (father of Chief Joseph) was among several chiefs of the Nez Perce bands that signed a treaty that established the first reservation boundaries for the Nez Perce which included practically all traditional lands. However, later in 1863, the U.S. Government decided to establish a new treaty to overrule the previous one signed in 1855. Old Joseph did not sign this new treaty, which cut the land size down to about 1/10th of what was promised to the Nez Perce. This also excluded the Wallowa Valley, home to Old Joseph's (and later Chief Joseph's) band of Nez Perce. Therefore, Old Joseph's band remained there. In 1873, President Grant formally set aside the Wallowa as a reservation for the roaming Nez Perce and whites were ordered to remain out of the area, much like in agreement with the original 1855 treaty.
Unfortunately, due to several circumstances (discovery of gold, encroachment of expanding whites), President Grant rescinded his previous edict and reopened the Wallowa to white homesteaders in 1875.
Old Joseph and Chief Joseph continually appealed to federal agents to resolve the conflicts occurring over their homeland. For a time, the government decided not to do anything about it until harassment continued. Eventually, the government realized that one side could respond with violence. The government, of course, sided with the white settlers and referenced the 1863 treaty as to why they were allowed to enter the land. Much negotiation happened and it ended with neither side really giving up. This all contributed to a build up of tensions between the two parties.
Just prior to war
Old Joseph passed away in 1871, leaving his band to Chief Joseph. While he was given counsel to stay on his lands in the Wallowa Valley, Joseph concluded that they could not resist the U.S. military who were now threatening to move all Nez Perce bands not on the reservation by force. While many disagreed, Joseph and his brother Ollokot convinced their bands to head to the reservation.
However, as you mentioned, a raid occurred in June of 1877. Hostilities and tension were running high. Several Indians attacked a group of 4 white men and then proceeded to attack another group, killing up to 15 additional settlers.
Beginning of the war
Many of the non-treaty Nez Perce bands were now actively against moving to the reservation. Many were now crying for war. Bands began departing to hiding places further south. Chief Joseph decided not to abandon his people and joined his band in White Bird Canyon. At this time and during their retreat, several other bands joined them.
Once General O. Howard heard news of the raid, he mobilized his forces from Lapwai and went to round up the non-treaty bands to, as per his original mission, force them onto the reservation.
The first battle occurred at the Nez Perce village in White Bird Canyon. Blood was now spilled.
While the Nez Perce are famous for their retreat, it wasn't a retreat in the sense that we were trying to avoid fighting. It was a tactical retreat. This was very much a war. But you are correct that the Nez Perce were generally passive toward noncombatants. This was not a war they originally wanted and exhausted every possible method for peace.
Decisions during the war
Many of the warriors and chiefs had grown tired of the war fairly quickly by the end of July in 1877. It was decided that they would keep withdrawing and head towards the Crow tribe in Montana, a suggestion made by Chief Looking Glass. They believed that if they could cross the mountains, General Howard would be satisfied, give up, and the war would be over. This is not what happened, though.
Under increasing criticism from Washington, the press, and the public, General Howard committed himself to following the Nez Perce and completing his mission. This is most likely a primary reason for why the chase continued. The public wanted revenge for the deaths of the white people and Howard's reputation was now on the line. After losing several battles and allowing the Nez Perce to make it as far as they did, why let them continue?
After being betrayed by the Crows, the Nez Perce bands decided to head to Canada and join Sitting Bull. While I cannot say for certain, the U.S. military forces more than likely realized they were trying to escape to Canada once they were about 30 miles outside the border. Crossing the border would mean they would be free and would cause a great embarrassment for the military.
Aftermath
So what can we conclude? The main reason General Howard gave pursuit for so long was because of pressure, both from the public and the government, to force the Indians onto the reservation, to avenge the deaths of white settlers, and to make up for the humiliating losses in battle they already suffered. While I am uncertain, he more than likely knew they were headed for Canada and having them escape would look pretty terrible combined his already failed attempts. To back this up, we can look at the punishment that Chief Joseph and his band received following the war. They were sent down to Oklahoma - Indian Territory - and not permitted to return to either the Wallowa Valley or the reservation in Idaho. After some time, they were shipped up to the Colville Reservation in Washington State where Chief Joseph died. The government had no desire to live up to their treaty of 1855 or honor the words of General Miles and send Joseph back to the new reservation.
One could argue as well that the U.S. Government feared the Nez Perce escaping to Canada and linking up with other tribes, including Sitting Bull's group, who were actively avoiding surrender at that time.
Edit: Sources are Chief Joseph's People and Their War and The Nez Perce Indians and the Opening of the Northwest, both by Alvin M. Josephy, Jr.