Meet the Shannons! In a dystopian future on a burned-out earth where people need "rebreathers" just to breathe the polluted air, they've defied the billboards that proclaim "Overpopulation equals extinction!" and "A family is four!" Treasure equals an orange that is almost freakishly colorful amidst the drab backdrop. Zoe is the moppet who the Shannons aren't allowed to have, and when the population police come calling, the Shannons stick her in an air vent and hope for the best. But babies cry, they always do, and when Dad goes berserk and attacks the police, he winds up in prison.
Two years later, and presumably many labored-breathing sodomizations later, Elizabeth (a doctor) visits her husband in a prison so ridiculous that she's able to slip him a laser that can cut through anything, which he'll need in order to break out, rescue Zoe, and join his wife and Maddie and Josh as they've been selected for the Tenth Pilgrimage, a one-way trip through a "time fracture" back 86 million years ago. Zoe can't come, because that would be rewarding the family for "bad behavior," which raises the question of why they're allowed to leave this polluted world in the first place.
We barely see any of the escape or rescue (John is able to break in to "Hope Plaza" by strapping himself to the underside of an armored vehicle. I guess in the 22nd century, they've forgotten all about mirrors on sticks). An accomplice has a backpack waiting for him, and John manages to rejoin his family (Josh bids a strangely non-tearful goodbye forever with his girlfriend) as they trudge toward the Stargate or time fracture or whatever the hell, but he gets pulled out of line and questioned about being seen leaving a restricted area. The family is reluctant to go through the time rift without him, but Elizabeth tells her daughter, "He'll find a way! He always does!" Well, except for the time he punched police and wound up in jail for the next two years and only broke out after you brought him a laser slicer.
Turns out "punching police" is his plan here too, and he punches his way through the authorities and into the time rift and 86 million years into the past, full of fresh air and trees. The weapon he's got doesn't sit well with the authorities on the other side, and they decide to STAB HIS BACKPACK, because of course that's what you would do, and you're nuts if you think this show is killing a kid this early. The family manages to prevent the backpack-stabbing, and when it's opened we find, to no one's surprise, Zoe.