Another nesting question

I have a function F[i_Integer,B_List,l_list] which spits out a list {B',l'} where B' and l' are of the same dimension as B and l, respectively. Now I want to get F[j,B',l'] and I then want to use the output of that again to get F[k,B'',l''], etc. (note that the integer argument changes at each step). What is the best way to do this? I’d be grateful for any help.

F[i_Integer, B_List, l_List] := {B - i, l + i}

B = Array[b, 3]; l = Array[L, 4];
SeedRandom[123]; {i, j, k} = RandomInteger[{-5, 5}, 3]

(* {2, -1, 5} *)

Fold[F[#2, Sequence @@ #1] &, {B, l}, {i, j, k}]

(* {{-6 + b[1], -6 + b[2], -6 + b[3]}, {6 + L[1], 6 + L[2], 6 + L[3], 
  6 + L[4]}} *)