Which are some xbox 360 games like GTA IV or Saints row or other sandbox games with the same kind of missions?

 GTA IV and Saints Row2 are at the top of the list for crime related sandbox Games.They influenced a new genre of sand box open world style games.

Just Cause for xbox360 offers the freedom to tackle your assignment however you want: play the island's factions off against one another; incite a rebellion among the masses; or build alliances with rebel forces and drug cartels. Explore the game's world of 1,225 square miles of mountains, jungles, beaches,cities and villages with a variety of air, land and sea craft.

  Prototype for XBOX360 Game Features:

    Fast & Deadly Shape-shifting Combat: Reconfigure your body to the situation at hand…from Claws to Blades to Hammers to Whips…choose the right weapon for the situation. Change to shield or armor for defense or use advanced sensory powers (thermal vision, infected vision) to track your enemies.
    Over-The-Top Locomotion & Agility: Seamlessly and fluidly bound from building to building, run up walls, bounce off cars and everything in your path. Adaptive parkour lets you move freely through the open-world environments of New York city.
   Unique Disguising Abilities: Consume anyone at anytime, take on their appearance and assume their memories and special abilities.
     Deep, Conspiracy Driven Storyline: Wake up with no memory of the past…just mysterious powers and a link to a town in Idaho. Delve into the mysteries of your origin, the true nature of your power and your part in a conspiracy 40 years in the making.


In Crackdown  for Xbox360 you must bring order to a corrupt city by defeating gangs, using superhuman power given to you by an Agency. You can also throw cars.
Crackdown pushes the action-driving hybrid genre into the next generation with the first ever truly 3-D playground. Enforce justice by any means necessary in a crime-ridden urban center built to encourage the exploration of the full width, depth, and height of the city. Coupled with highly innovative co-op gameplay, a first for theg enre, and an interactive world where nearly anything can be used as a weapon, create a volatile cocktail of action as you clean up the streets.