I was a bit surprised to find out recently that the famous Nubian pyramids at places like Meroe were much younger than I had assumed, first being constructed in the 8th century BC, and continuing well into the first millennium AD. The iconic Pyramids at Giza predate them by well over a thousand years, it seems akin to somebody today building a Roman Tomb for themselves.
So I'm curious, were the Nubian pyramids necessarily a direct imitation of the old Egyptian pyramids, or were they a tradition that had developed more locally to Nubia but just happened to resemble the Egyptian pyramids? If they were deliberately drawing a parallel, why were people in Nubia resurrecting an already ancient tradition that had long since remained moribund in Egypt itself?
The short answer is that they are imitations of the Egyptian pyramids, and were built to deliberately revive a "dead" tradition. (Though it is worth mentioning that it did not die out after the Giza complex was built during the Old Kingdom; smaller pyramids were built throughout the Middle Kingdom, until c. 1650 BC).
Between 830 and 750 BC, Egypt effectively split into a number of smaller states, with independent kings ruling from Thebes, Herakleopolis, and several cities in the Nile Delta. Some of these kings were native Egyptians; others, especially in the Delta, were invaders from Libya. Taking advantage of this unstable situation, the Kushite ruler Kashta began to assert control over Upper Egypt. Beginning in 744 BC, his successors Piy and Shabako led campaigns northwards along the Nile, ultimately conquering Egypt and establishing a new line of pharaohs (the 25th Dynasty).
To help establish themselves as the legitimate rulers of Egypt, the Kushite kings adopted the symbols and religion of the Egpytian pharaohs. They were especially interested in building up associations with the Old Kingdom, which the Egyptians themselves viewed with nostalgia as the first great era of their civilization. The 25th Dynasty consequently took up residence at Memphis, the Old Kingdom capital, and funded construction projects that imitated Old Kingdom styles. This included the revival of pyramid-building, though the pyramids were built not in Egypt proper but at the Kushite royal cemetery, El-Kurru, outside the Nubian capital city of Napata.
This was not a drastic step for the Kushite rulers to take. They were "Egyptianized" already, due to Egypt's past dominance over their homeland. Egyptian religion, for example, was well-established in Nubia; when the region was under Egyptian control in the 13th century BC, a large temple to Amun was built at Jebel Barkal, 20 km from El-Kurru. That said, pyramids were not a traditional feature of Kushite royal burials. The oldest Nubian pyramid is presumed to be the tomb of Kashta, who had taken the first steps towards the conquest of Egypt.
The 25th Dynasty ended when Assyria invaded Egypt in 663 BC. The Kushite kings were forced to gradually retreat, first to Napata, and later to Meroe after Egypt regained its independence and began to push southwards along the Nile again. At Meroe, they maintained some of the Egyptian customs that they had adopted, including pyramid-building. However, with the direct link to Egypt lost, many of the details began to develop in a new direction. In particular, there was a rapid transition from writing inscriptions in "literary" Egyptian (using hieroglyphic), to using simpler or ungrammatical Egyptian, and finally to using the Meroitic language.
Sources: mainly, John Taylor's chapter on the Third Intermediate Period in The Oxford History of Ancient Egypt (ed. Ian Shaw). Some details from Alan Gardiner, The Egyptians.