In Bavli Sanhedrin 32b it says:
"The Sages taught: The verse states: “Justice, justice, shall you follow.” This teaches that one should follow the best, most prestigious, court of the generation. For example, follow after Rabbi Eliezer to Lod, after Rabban Yoḥanan ben Zakkai to Beror Ḥayil.
The Sages taught: If one produces the sound of a millstone in the city called Burni, this is tantamount to announcing: Week of the son, week of the son. If one displays the light of a lamp in the city called Beror Ḥayil, this is tantamount to announcing: There is a wedding feast there, there is a wedding feast there.
It seems to me that this anecdote about Burni and Bror Chayil is brought in relation to RYb"Z moving to Bror Chayil. However, other than the common place-name, there doesn't seem to be any actual connection between the two: It seems RYb"Z moved to Bror Chayil after spending some time in Yavneh rebuilding the nation, post-destruction. But adding the anecodte about Bror Chayil seems really random. The braita about Bror Chayil and Burni is compared by pretty much all of the commentators on Sefaria (as well as many other sources that I found) with the Yerushalmi Ketubot 1:5:
"In earlier times they decided on a persecution in Judea...They decreed that her husband should come to her when still in her father’s house; for when she knows that her husband’s fear is on her she is drawn after him...What kind of a sign did they have? The talk of a cook in town: There is a wedding meal, there is a wedding meal; there is light in Beror-Ḥayil: a week for a son, a week for a son. When the persecution ended, the custom did not end. Rebbi Hoshaia’s daughter-in-law was pregnant when she married definitively [according to Pnei Moshe and Korban Ha'edah she arrived at her chuppah already pregnant; Rabbi Hoshayah was one of the last tannaim - this shows the custom lasted at least that long]."
Most commentators also point out that that the Bavli, other than mentioning Burni (which the Yerushalmi doesn't), switched the order - in Bror Chayil a light signifies a "week of a son" instead of a wedding feast.
But another parallel is 17th of Elul in Megillat Taanit where it is stated that the this poem or song originated with the anti-Torah laws from the time of the Greeks, centuries prior to RYb"Z's time.
So what does this braita actually have to do with the previous one?
I think it's possible that the fact that it was placed here hints towards a special situation in Bror Chayil during RYb"Z's time, and that's why there are differences between the Bavli's poem and the versions in Megillat Taanit and the Yerushalmi. But I haven't been able to figure out what that special situation might be.
Does anyone know (or can think of an explanation)? Or is there any other way to explain the connection between the Braitot?