It's a definite possibility. However, would this be done in a completely philanthropic manner? Likely not.

As an example, the Nomarch Ankhtify of the First Intermediate Period (2160-2055 BC) has inscribed his efforts to combat famine in his tomb not too far from the city of Thebes.

There he writes: "Horus brought me to the nome of Edfu for life, prosperity, health, to reestablish it, and I did (it). For Horus wished it to be reestablished, because he brought me to it to reestablish it. I found the House of Khuu (Edfu) inundated like a marsh, abandoned by him who belonged to it, in the grip of a rebel, under the control of a wretch. I made a man embrace the slayer of his father, the slayer of his brother, so as to reestablish the nome of Edfu.....I was the beginning and the end of mankind, since nobody like myself existed before nor will he exist; nobody like me was ever born nor will he be born. I surpassed the feats of the ancestors, and coming generations will not be able to equal me in any of my feats within this million of years. I gave bread to the hungry and clothing to the naked; I anointed those who had no cosmetic oil; I gave sandals to the barefooted; I gave a wife to him who had no wife. I took care of the towns of Hefat and Hor-mer in every [situation of crisis when] the sky was clouded and the earth [...] of hunger on this sandbank of Apophis. The south came with its people and the north with its children; they brought the finest oil in exchange for the barley which was given them. My barley went upstream until it reached lower Nubia and downstream until it reached the Abydene nome. All of Upper Egypt was dying of hunger and people were eating their children, but I did not allow anyone to die of hunger in this nome."

You can see the act of "kindness" was more intended to bolster his own reputation in his quest for power.

He had taken over a fellow Nomarch's land and wanted to strengthen his grip over the people. The best way to do that was to give the people the food they needed during this famine.

Keeping a private grain stock allowed men like Ankhtify to consolidate their power. How does this effect the refugee question? Well Egypt was quite decentralized in this period and Edfu was under a seperate rulership from Ankhtify's. In order to grow stronger, he conquered Edfu and (according to his inscriptions) fed the populace. In this time, almost nothing would stop a family from sneaking into a Nomarch's land and buying a share of the grain.

If perhaps foreign aid seems to far of a jump to make from this, note that in the New Kingdom, King Merneptah sent a shipment of grain to the King of Ugarit to "relieve the famine in Ugarit." This was a much more globalized time, but in the time period of Ankhtify, border control wouldn't have been as strong as the later Middle/New Kingdom Fortresses allowed it to be.

The last, and possibly most famous example would be the legendary Sea Peoples. They likely were escaping famine in the Aegean when they decided to settle in Egypt and Egyptian held lands. While the fierce resistance to their landings in Egypt is well recorded, there are spots that the archaeological record does support a peaceful settlement on former Egyptian imperial lands.

So, it can't be said conclusively what their policy on refugees was, but from the few accounts we have we can see that people did indeed move into Egypt at various times of its history. Not only that, but the Egyptians with their usual abundance of food could use their grain as a tool to consolidate power. In other words, they seem to have been willing to share.

(Some good sources include: Ancient Egyptian Literature Vol 1 by Miriam Lichtheim, and 1177 BC by Eric Cline. And I ALWAYS recommend picking up Anatomy of a Civilization by Barry Kemp. You'll see that book in practically every Egyptologist's bibliography.)