My concept.
A thermonuclear orion departs for another star. It carries 200 colonists (100 males/100 females) and around 20,000 metric tons of equipment and supplies. Traveling at .2c, it would take 100 years to reach Delta Pavonis. Nine more identical ships depart for the same destination, launching ten years apart. This would be a total of 2,000 colonists and 200,000 metric tons of equipment and supplies. If all goes well, the population should be well over 20,000 inhabitants by the time the last ship arrives. By launching them in intervals of ten years, it would eliminate the cultural separation that you mentioned.