The Caucasus is rarely the whole trip. Most people arrive from somewhere and leave for somewhere else, and the three capitals are better connected than their size suggests.
From roughly November the direction of travel reverses. Tbilisi, Yerevan and Baku all sit within four hours' flying of the Egyptian coast, and the fares collapse once the mountain season ends. Cairo, Hurghada and Sharm el-Sheikh are the three names you will see on the departure boards most often, and for a region with long grey winters that is not an accident.
Egypt runs on different rules from anywhere in the Caucasus, and the gap catches people out. Alcohol is sold, but not everywhere. Photography near anything official is a bad idea. And gambling, which is openly legal, is legal in a way that surprises most arrivals: licensed casinos operate only inside hotels, entry requires a foreign passport, and Egyptian citizens are barred from them by law.
That split — legal for visitors, closed to residents — is why the Egyptian market looks so strange from outside, and why the useful reading on it is local. If that is the part of the trip you are researching, a current rundown of casino sites egypt will tell you more than a general travel page can.
Istanbul remains the cheapest hub out of all three capitals, and the Gulf carriers run deep discounts through the shoulder months. Overland, the only routes that stay open year-round run west into Turkey and north through the Caucasus passes — see routes for how those connect.