To determine which geographic feature escaping enslaved people would need to cross to gain their freedom on the Underground Railroad, it is essential to consider the route taken by many individuals seeking freedom.
Typically, escaping enslaved people traveled northward to reach free states or Canada. Of the rivers listed — the Red River, the Brazos River, the Colorado River, and the Rio Grande — the most significant river that would fit this context is the Rio Grande.
The Rio Grande forms part of the border between the United States and Mexico, and crossing it could symbolize a journey towards freedom, especially for those escaping from southern states.
So, the correct answer would be the Rio Grande.