I'll post the "cliffnotes version", I"m sure you'll get much more detailed responses as well.
A Baladi Progression is a type of song comprised of three basic parts -
1) taxim
2) taxim progressing with tabla accents
3) the accents progress into rhythms and the song continues, often building up to a fairly fast pace.
It is very Egyptian and classically a part of an Egyptian style set. If you watch full shows from almost any of the Egyptian dancers from the 70s and 80s you'll see it.
Here is an example:
www.youtube.com/watch
In Egyptian shows a lot of the songs start with taqsims- Om Kolsoum songs will, Saidi often will, so it's important to be able to hear the difference between a beladi progression and an actual song in it's instrumental form.
another example of the progression
www.youtube.com/watch

Beladi of course also is the common American name for a rhythm (Masmoodi Saghiir) and a term that roughly means "country" (as in nation), and also "country" as in from the country or having a country/old time/earthy/"of the common people" feel.
Mostafa Sax has a lot of lovely examples of beladi progressions- Many of his would have to be edited to use for shows- as they tend to meander a long time and fade out.
beledi, belady, baladi...look up all sorts of different spellings. A lot of stuff on youtube is mislabeled, so be careful.

Thank you so much Samira. I myself posted once what I found on a webpage - yours was easier to understand, but since you posted the Najua clip also this scheme seemed more understandable.
1. Awwady (swaying like an ood/bamboo-cane), slow improvised taqasim with small moves full of feeling. There is the musical style Mawal as free nostalgic singing without rhythm. The dancer slowly "sways" herself into the music.
2. Me-Attaa (broken pieces of music and rhythm), introduction to the rhythm, with the tabla the play of question and answer starts, the dancing style yet rather conservative and shy, but sensual and feminine.
3. Maqsoum: the rhythm is established, goes faster, dancer dances more extrovert, hipwork!
4. Tet (4/4 rhythm accent on 2 and 4 with "tiit toot tet teeit" played by the mizmar), nostalgic sound of the Egyptian countryfolks
5. "coming home", transition to fellahi-rhythm, danced with Egyptian walk
6. Awwady Taqasim as calm end, or "cooling down" (f. ex. "Lucy from Kairo" out of the album best of baladi & saidi by Hossam Ramzy)
a drum solo can also be added. "

Now thank you so much for the links. Now I always wanted to see a piece that has all this, to understand how the classical balady goes.
1,2,3 and 6 I can identify in the Najua-clip, yes, plus the drum solo. Is there the tet and the fellahi too? Are they a must or could they drop out?
Yesterday I came across a clip on bhuz - so nice: youtube.com/watch
(This is just on maksoum, no progression)
.

Shareen calls it Beledi-Taqsim, it's one of her basic topics - it's on the schedule for her workshop in VA.