First, in the books, The Golden Compass, The Subtle Knife, and The Amber Spyglass, they state that there are many worlds and one that is like the crossroad world.
Now we get to the refrence to the Chaos Emeralds. Because they are pure energy, when Sonic or Shadow hold them, they absorb some of the power.
First, the chaos emerald shoots a beam of energy that opens a portal to the crossroad world. Then, they pull the person holding them through, close the portal, and start their molicules to rotate very fast, to match the Emeralds speed. Then, because their molicules are moving so fast, the chaos emerald, and the person holding it, can pass through the solid objects in the crossroad world. Finally, they meet the destination in the crossroad world where the Emerald cuts open a portal to the world that the person left and pulls the person through and then finally closes the portal.
All of this happens in an instant.
Finally, the sparkle thing that appears before and after a person uses chaos control is the emerald cutting open and closing the portal.