Remove the center stems from all the greens and slice the leaves into 1/2-inch ribbons. Pour the olive oil into a large Dutch oven set over medium-high heat. Once hot, add the onion and garlic and saute until tender and fragrant, about 4 minutes. Season with salt and pepper.
Stir in the ribbons of mustard greens, turnip greens and kale in batches, adding the next batch as the one prior wilts down. Once those three greens are added to the pot, pour in the broth and cook 15 minutes. Then add the ribbons of Swiss chard and cover with a lid. Let simmer 5 more minutes. Taste for seasoning. Spoon the greens into a large serving dish.