Migration Routes for Animals from Asia to America
The jaguar is a close relative of the Asiatic leopard and must have had a common ancestor within the last 5 million years. The South American tapir is obviously closely related to the Malayan tapir and must also have had a common ancestor within he last 5 million years.


While the first statement is more or less correct, the latter is not. The Asian and American branches of the tapirs split about 25 million years ago.

Regarding migration routes, felines have migrated back and forth between Asia and North America many times in the last eight million years or so. The ancestors of Tapirs apparently migrated back and forth between North America and Europe before 25 million years ago, when the climate was much warmer and the continents had slightly different shapes.


  ... is there a possibility that when Africa and Amazonia were closer together, immigration from Africa could have taken place?


Regarding South America, there were very few placental mammals in South America until about 3 million years ago, when the Isthmus of Panama formed. The age of the Isthmus is debated. The sudden appearance of placental mammal fossils about 2.7 million years ago in South America, including jaguars and tapirs, is not. South America's jaguars and tapirs migrated to South America via North America rather than Africa.