I saw this mentioned in a Chabad chinuch document (i.e., teach your kids to get dressed from the top down, etc.). I also saw it mentioned here. What is the source of this custom, and who follows it?
Since you mentioned chabad, I decided to check the shulchan aruch harav. See below pulled from a site entitled by that very name.
Which clothing is one to wear first?
The custom amongst Chassidim and G-d fearing Jews is to get dressed from top to bottom. Thus one is to first wear the Yarmulke, then the undershirt, Tzitzis, shirt and then the underwear, pants, socks and shoes. The custom amongst Ashkenazi Jewry is to get dressed from bottom to top, first putting on the pants and only then the shirt and Tzitzis.
Which clothing is one to remove first?
The custom amongst Chassidim and G-d fearing Jews is to get undressed from down to up. Thus one is to first remove his pants and then his shirt and Tzitzis [and last he is to remove his Yarmulke]. The custom amongst Ashkenazi Jewry is to get undressed from top to bottom.
The footnotes which I shall quote below quote the sources and give an explanation, in some detail.
 See Toras Yekusiel 1/8 and 101 for a thorough discussion on this matter.
The Chassidim and Ashkenazim practice different orders in dress. The Chassidim get dressed starting from the top, while the Ashkenazim from the bottom. The order of getting dressed and undressed is discussed in Tractate Derech Eretz Zuta and Raba. It is also mentioned with regards to how the Kohanim would dress in their priestly garb when serving in the Temple. In Derech Eretz Zuta 8 it states to get undressed from bottom to top [and hence one is to get dressed from top to bottom]. In however Derech Eretz 10 it implies one is to get undressed from top to bottom, and so is the order of dress of the Kohanim. The following are the opinions and reasons behind the customs.
 Miseches Derech Eretz Zuta chapter 8; Oar Tzaddikim 1/8; Toras Yekusiel 1/8 and 101; Mishneh Sachir 2/2; Minchas Yitzchak 4/60; Shulchan Hatahor Tznius 4; Kaneh Bosem 2/1; Shraga Hameir 1/24; Piskeiy Teshuvos 2/4
The source: In Derech Eretz Zuta 8 it states to get undressed from bottom to top [and hence one is to get dressed from top to bottom]. The Sefer Oar Tzaddikim ibid of Rav Meir Papirish, a student of the Arizal, writes to precede the Tzitzis to the pants. The Minchas Yitzchak ibid writes this order as a tradition from the Tzaddik of Ziditchov. He then concludes that this is the “custom of all the G-d fearing”; The Shulchan Hatahor ibid writes that those that get dressed and undressed the opposite way are considered Perutzim [licentious].
The reason: This order is followed in order so one precedes his Tzitzis which is a Mitzvah to other clothing. [Oar Tzaddikim ibid; Toras Yekusiel ibid]
 Ben Ish Chaiy Vayishlach 16; Kaf Hachaim 2/9; Minchas Yitzchak 4/60
 Toras Yekusiel ibid based on the order mentioned in Derech Eretz 10, and the order of dress and undress of the Kohanim mentioned in Yuma 23. The Kaneh Bosem ibid writes that the Ashkenazim misunderstood the wording in Derech Eretz 10 and in truth it is a source for the Chassidic custom and not the Ashkenazi custom, and so retracts also the Toras Yekusiel in a latter response that he made a mistake in the understanding. The dispute stands as with regards to if the word Pirkoso is understood to mean pants or shirt. Since it is the last item mentioned there to remove, therefore if it’s translated as pants it is a source for the Ashkenazim. If it is translated as shirt then it is a source for the Chassidim.
The reason: This order is followed in order so one covers his Erva as soon as possible. [Toras Yekusiel ibid]
 See previous footnotes
 Toras Yekusiel 1/8 and 101; Mishneh Sachir 2/2; Minchas Yitzchak 4/60: “custom of all the G-d fearing”; Shulchan Hatahor Tznius 4; Kaneh Bosem 2/1; Piskeiy Teshuvos 2/4
 Minchas Yitzchak 4/60; Piskeiy Teshuvos 2/12; See Shaareiy Teshuvah 2/3; Birkeiy Yosef 2
 Toras Yekusiel ibid.