To unlock the "HAPPY FAMILY" trophy in Detroit: Become Human, you must make sure Kara Alice and Luther all reach Canada. It's unclear at this point whether all three can survive the river crossing in any scenario so I suggest playing it safe by going through border control. Besides, you can only get the below trophy for an official border control crossing:
Here is one example branch of the flowchart "BATTLE FOR DETROIT - KARA LEAVING DETROIT" to get Kara, Alice and Luther across the border, assuming you've managed to get Kara and Alice this far and Luther was saved during "CROSSROADS".
Bear in mind that this is one of perhaps many possible routes to getting Kara, Alice and Luther through border security. I will update with other options as they are discovered.
*** Spoiler - click to reveal ***
It's worth noting that Markus was conducting a peaceful demonstration at this point and the public are sympathetic. Not sure if that impacts the outcomes.
1. Successfully hide from the military until you see Luther & Jerry being held captive. Sneak around the car, pick up the brick and knock out the guard looking over them.
2. Choose to approach the checkpoint rather than take a detour.
3. Choose "ALICE EXCUSE" when talking to the guard. Either "COLD" or "FEAR" work as answers when the guard wonders why Alice is shaking.
4. When the guard calls you back, do not draw your gun at either opportunity given. He will only hand Alice her lost glove.
5. When you get to the bus depot, head straight over to the woman and baby on a bench to the left. Eventually the family will drop their bus ticket. Steal it and don't return it, just get on the bus.
6. At border control, look around and make sure to connect to Jerry in another line. He will offer to sacrifice himself. Take him up on his offer when you reach the desk.
If you get Kara, Alice and Luther into Canada by the game's closing credits, you will get the trophy when you reach the flowchart screen.