A consequence of the geometric torsion conjecture for abelian varieties over function fields is the following. Let k be an algebraically closed field of characteristic 0. For any integers d,g⩾0d,g⩾0 there exists an integer N:=N(k,d,g)⩾1N:=N(k,d,g)⩾1 such that for any function field L/kL/k with transcendence degree 1 and genus ⩽g and any d-dimensional abelian variety A→LA→L containing no nontrivial k-isotrivial abelian subvariety, Ators(L)⊂A[N]A(L)tors⊂A[N]. In this paper, we deal with a weak variant of this statement, where A→LA→L runs only over abelian varieties obtained from a fixed (d-dimensional) abelian variety by base change. More precisely, let K/kK/k be a function field with transcendence degree 1 and A→KA→K an abelian variety containing no nontrivial k-isotrivial abelian subvariety. Then we show that if K has genus ⩾1 or if A→KA→K has semistable reduction over all but possibly one place, then, for any integer g⩾0g⩾0, there exists an integer N:=N(A,g)⩾1N:=N(A,g)⩾1 such that for any finite extension L/KL/K with genus ⩽g, Ators(L)⊂A[N]A(L)tors⊂A[N]. Previous works of the authors show that this holds--without any restriction on K--for the ℓ-primary torsion (with ℓ a fixed prime). So, it is enough to prove that there exists an integer N:=N(A,g)⩾1N:=N(A,g)⩾1 such that for any finite extension L/KL/K with genus ⩽g, the prime divisors of |Ators(L)||A(L)tors| are all ⩽N.