There are 4 different off-road missions in Liberty City, 1 in Portland, 2 on Staunton Island and 1 on Shoreside Vale. You trigger them by jumping in the cars you use for the missions. The missions are time-limited checkpoint races. The timer starts when you drive over the first checkpoint and all the checkpoints can be collected in any order you want. The trigger cars are marked on the maps with this icon:

Mission 1: Patriot Playground
Location: Portland View, Portland
Reward: $30.000
Objective: Collect 15 checkpoints in 5 minutes. Each checkpoint adds 20 seconds to the timer.
How to start mission: You will find a Patriot in the parking lot at Supa Saves.

Mission 2: Multi-Story Mayhem
Location: Newport, Stanton Island
Reward: $30.000
Objective: Collect 20 checkpoints in 2 minutes.
How to start the mission: Park a Stallion in front of the garage, get out of it and the get back into it to start the mission.

Mission 3: A Ride in the Park
Location: Belleville Park, Staunton Island
Reward: $30.000
Objective: Collect 12 checkpoints in 2 minutes. Each checkpoint adds 10 seconds to the timer.
How to start the mission: You will find a Landstalker inside the park.

Mission 4: Gripped!
Location: Shoreside Vale
Reward: $30.000
Objective: Collect 20 checkpoints in 5 minutes. Each checkpoint adds 15 seconds to the timer.
How to start the mission: You will find a Patriot in the pic-nic area.