[bsamuels]

At the level of technological sophistication required to send a probe to another star system, you could probably sequence/modify the genome of the eggs/sperm to produce humans that have a lust for exploration and pioneering.

Now it's just a question of whether genome modification is ethical.

[JoeAltmaier]

Not quite: the 20 generations that are born and die on the ship don't get to explore anything. You'd want people content to live in a crowded can.

[AOsborn]

In a related point - what would the cultural mindset of the final descendants become, both individually and collectively?

They would have no shared experiences with those who first set off.

Ideals could be passed down, but with cultural evolution through the generations, would their priorities or preferences even be remotely similar?

[JoeAltmaier]

Many science fiction stories written about this. The most popular theme is the breakdown of civilization and an ignorant crowd of savages set down on the destination world. Other authors have them undergoing violent revolution a dozen times over the centuries, recapitulating human history. The resulting world is populated by whatever form of government is extant at the time of arrival. Still others imagine them passing serenely by the destination, no longer interested in their progenitors plans and seeing no other future for themselves than perpetual ship-board life.