If "the mission" is survive, then yeah probably. I doubt the colonists would hold any loyalty to Earth, but they'ed definitely set up a colony for their own sak

If "the mission" is survive, then yeah probably. I doubt the colonists would hold any loyalty to Earth, but they'ed definitely set up a colony for their own sake. Also we do have plenty of institutions which have to some degree or another stuck to the mission many lifetimes after their founding. Religious institutions like the catholic church come to mind - obviously much has changed and plenty would argue about how well current behavior matches the founders' intent, but thousands of years later there is still an organization of people working towards a broadly similar goal. Less controversial examples include some construction projects which took centuries, like the Cologne Cathedral. It should be noted that while none of the original crew would survive the journey, there would be an unbroken chain of people raising new crew members, educating them to the mission, and their adoption of the existing crew's values as their own, and propagating them forward, would be necessary for both their own well being and the group's. There would be little if any outside influence to cause the group to diverge from its mission, and no realistic alternative they would be able to pursue even if they wanted to. There might be ideological drift, but it would probably be a lot less than people who have been free to do whatever they want for the same period of time.