Heat the olive oil in a large skillet over medium-high heat. Add the pork shoulder chunks and brown on all sides for about 6 minutes total. Transfer the pork to the slow cooker.
In the same skillet, cook the onion for 4 minutes until it softens. Add the garlic, chipotle peppers, and cumin. Stir for 1 minute, then scrape everything into the slow cooker.
Pour the chicken broth, black beans, corn, and diced tomatoes into the slow cooker. Stir to combine. Season with salt and pepper.
Cover and cook on low for 8 hours, until the pork is very tender. Stir once halfway through if possible.
Taste and adjust salt if needed. Ladle into bowls and top with cilantro. Serve with lime wedges on the side.