I'm Peruvian and I spent the first 13 years of my life there. I don't know know about Mexico, but I do consider Peru to be a the successor of the Inca Empire. If you see a map of the tawantinsuyu, you will see the empire overtook almost all of Peru, Ecuador, half of Boliva and Chile, and even a little bit of Argentina and Colombia.

However, The Inca empire was never a unified nation. Besides having one ruler, the Inca empire was mainly made up of different tribes each with their own language, culture, and political structure. Many of the tribes were not in accords with the Inca, and so tribes would constantly emancipate from the his ruling, while others would join. The only permanent and constant variable in the Empire was the capital of Cuzco - a city that is still standing today. In fact, a visit to Cuzco today will show you that not only the government but the people as well proudly display the Inca rainbow flag in public, the same flag used by the Incas. It can also be said that the core, original, and early territory of the Empire took place almost exclusively within Peruvian boarders, later expanding north and south.

Of course one could also make the argument that there is really nothing to pass on since the Inca Empire officially died with the Conquistadors, but unlike in North America where tribes were relocated, assimilated, and killed off, in Peru tribes were allowed to keep their own traditions in a sythesis of Catholicism and Inca spirituality, which has played a large role of Peruvian identity.

I guess that same thing can be said for Bolivia, though Peru has much more historical grounds to claim to be the successor.

edit: Grammar