They do . . . but its not a strict carry over. The Dragoon striker battalions, and apparently the organization carries over to the line regiments- of which they only have 1 field force? with Alpha being a repple depple?- has the companies made of up 5 stars. Mech (Company Com/Majo or Capt) Star, 3 vehicle stars (1 star will have the vehicle stars commander, can be Lt but we mostly saw Sgt), BA star (Lt) is the organizational make up. Difference between Line & Strikers is the Line BNs have slower mechs & vehicles along with regular infantry in their mix.
By 3140 the Dragoons have . . .
Alpha Regiment
1st BN (Black Cats)- functional
2nd BN- training command
3rd BN- Personnel reserve
Gamma Regiment (45 mechs, 135 veh, 54 ASF, unknown BA/Inf)
Command formation?
1st BN- (15 mechs? 45 veh, unknown BA/Inf, 18 ASF)
2nd BN (same)
3rd BN (same)
Striker Battalions (Independent, 50 mechs, 135 veh, 54 ASF, 75 BA)
Striker Command Star (5 mechs)
Spider's Web
Tarantula (stood up 3129)
Wolfsbane (stood up 3139)
They are definitely a shadow of their former selves, and for all the Falcon salvage, access to Warden Wolf factories, access to Arc Royal MechWorks, Sea Fox merchants, and the Lyran supply chain . . . its a sorry state. They had to go challenge the Falcons for four Dropships 'because they needed them.'