Recently, I read a news article where an Egyptian guy is all like "We Egyptians are so awesome. We built the pyramids, y'know?". My first thought was that he was a bit foolish for claiming to be related to the ancient Egyptians. I'm Dutch myself and I know that during Napoleonic times the Dutch republic was renamed the Batavian Republic, after a tribe that lived in the Netherlands during Roman times. The implication being that the Dutch are descended from the Batavians, no doubt because of their supposed awesomeness. In modern times however, it's known that the Dutch didn't descend from the Batavians due to many migrations throughout the ages. It seemed obvious to me that this would be the case for Egypt as well.
However, the Wikipedia article on the Egyptian people seems to indicate that the Egyptians from ancient and modern times are indeed the same.
Judging from the Bronze Age collapse and the Great Migration period, for example, the lands neighboring Egypt have been resettled many times over, yet that very specific piece of land has been inhabited by the same ethnic group for... how many thousands of years? It seems to me this would make them a huge exception to the norm.
In modern times however, it's known that the Dutch didn't descend from the Batavians due to many migrations throughout the ages.
How is this known? Or rather, what are you sources that the modern inhabitants of the Netherlands are genetically discontinuous with their antecedents going back to the Neolithic? Your assumption is that historical migrations have had a major remodeling impact on populations of the Netherlands (and presumably the same must have occurred in Egypt).
You're not going to get very good answers anywhere. The only way to provide a definitive answer is to DNA sequence ancient Egyptian human remains (i.e., mummies). Unfortunately, while mummification preserves the form of bodies quite well, the hot climate of Egypt breaks down DNA quite rapidly. So unlike ancient human remains from colder climates in Europe, researchers have not had much success in sequencing ancient Egyptian DNA.
There was a paper from Nature a couple years ago about a new technique to sequence ancient Egyptian DNA, which appears more promising. It was kind of a proof-of-concept paper to validate methodology. Perhaps in a year or two we may see some papers using this method on some significant number of ancient Egyptian remains.
Other lines of evidence can come from either non-DNA sources (such as ABO blood groups, which preserve quite well in ancient Egyptian remains), or DNA sequencing of modern Egyptian populations. Paleoserological studies on ABO groups show that the ancient Egyptians match the current populations of Egypt quite well. However, they are not as definitive as DNA sequences.
DNA studies of modern Egyptians show that while Egypt went through several invasions in historic times, the DNA contribution from these invasions is only about 10%. The remaining 90% is much older, from neolithic times, presumably including the predynastic and dynastic periods. It consists of a mixture of northwestern African (Maghrebi), middle eastern, west Eurasian, Eritrean (Horn of Africa) and sub-Saharan African ancestries. The best we can tell, these components represent ancient admixtures, meaning they were already present in ancient Egypt.
So while the evidence isn't very firm, what we do have supports the idea that there is a continuity between ancient and modern Egyptians, with about a 10% foreign admixture from more recent invasions/migrations. The situation should become clearer in a few years as newer methods to sequence ancient Egyptian DNA are applied.
If by related you mean by blood (i.e. the degree to which modern Egyptians are thought to be descended from people who lived there 3000+ ya), it would be worth x-posting to /r/AskScience to get the latest state of genetic research on modern vs ancient populations (if any)
Meanwhile, a few posts in here that may be interesting