This is a great question! It's a side project I've been working on, actually.

Yes, we have evidence for Egyptians working abroad. One of the earliest instances is a literary example, the Tale of Sinuhe. In the story, Sinuhe is an Egyptian high official who traveled to the Levant after the king (Amenemhat I) was assassinated. While in the Levant, Sinuhe meets a ruler who has Egyptians working for him.

Then Ammunenshi, the ruler of Upper Retenu, took me to him, saying to me: "You will be happy with me; you will hear the language of Egypt." He said this because he knew my character and had heard of my skill, Egyptians who were with him having borne witness for me.

In The Tale of Wenamun, an Egyptian representative of the high priest of Amun named Wenamun travels to the Levant to acquire timber for a barque for the god's statue. At one point in the story, a local prince sends his Egyptian songstress to console Wenamun.

He sent his secretary out to me, bringing me two jugs of wine and a sheep. And sent me Tentne, an Egyptian songstress who was with him, saying: "Sing for him! Do not let his heart be anxious." And he sent to me, saying: "Eat, drink; do not let your heart be anxious. You shall hear what I will say tomorrow."

We also have requests from rulers abroad asking the Egyptian kings to send specialists abroad. For example, in one of the Amarna letters (EA 35), the king of Alašiya (Cyprus) wrote to the king of Egypt, either Amenhotep III or Akhenaten, to request an expert in augury.

Furthermore, my brother, the ox which my envoy has requested, give to me, my brother, and of the best oil, my brother, send me two kukubu jars, my brother, and send me one of the experts in vulture divination.

Egyptian physicians had an excellent reputation abroad and were highly in demand. The Hittite king Hattušili III wrote to his "brother" Ramesses II of Egypt requesting an Egyptian doctor to help his sister with her infertility issues. Ramesses II responded favorably but rather rudely.

Thus to my brother: (Concerning) what my brother has written to me regarding his sister Matanazi: 'May my brother send to me a man to prepare a medicine so that she may bear children.' So has my brother written. And so (I say) to my brother: See Matanazi, the sister of my brother, the king, your brother knows her. A fifty-year-old! Never! She's sixty! Look, a woman of fifty is old, to say nothing of a sixty-year-old! One can't produce medicine to enable her to bear children! Well, the Sun God and the Storm God may give a command, and the order which they give will then be carried out continually for the sister of my brother. And I, the king your brother, will send a competent incantation-priest and a competent doctor to assist her to produce children.

The Egyptians also sent builders abroad. One of the most interesting examples comes from the site of Der in Babylonia, where a local villager found a Kassite period (Late Bronze Age) brick containing a stamped inscription of King Kurigalzu II. The brick also contained a doodle of an Egyptian god and a solar disc in the Egyptian style, suggesting the presence of an Egyptian workman at the site.

Egyptians are well attested abroad in the Iron Age, particular in the Assyrian empire. Unlike in the Bronze Age, however, Egyptians were not being sent by their kings on missions abroad. Rather, it seems that enterprising Egyptians made their way to Assyria in order to engage in trade and commerce within the expanding empire.

Archive N31 from the city of Aššur included texts containing many Egyptian names, suggesting the inhabitants of the house were of Egyptian extraction.

An Assyrian text from Nineveh preserved the record of the purchase of a house by an Egyptian scribe.

A built house with its beams, doors and a yard in Nineveh, adjoining the house of Mannu-ki-ahhe, the house of Ilu-issiya, and the street —

Ṣilli-Aššur, Egyptian scribe, has contracted and bought it for one mina of silver (by the mina) of the king from Šarru-lu-dari, Atar-suru and Amat-Su'la, the wife of Bel-duri.

In another text, the son of an Egyptian sells his house to the chief chariot driver of the Assyrian king Aššurbanipal.

Seal of Issar-duri (and the) seal of Lu-šakin, son of Abšiešu the Egyptian, the owners of the house being sold.

In yet another text, the Egyptian Tab-Bel sells his female slave named Adimasia to a man named Adda-dimri.

An important but sadly fragmentary Assyrian personnel list contained the names of Egyptian scholars working for the palace.

[...]guršî

Ra'sî

Ṣihû

total, 3 Egyptian scholars

Huru

Nimmurau

Hur-waṣu

total, 3 Egyptian scribes

There are many other examples of people in Neo-Assyrian texts bearing Egyptian names or explicitly identified as being of Egyptian extraction, both before and after the conquest of Egypt by the Assyrians.