Let $M$ be a compact symplectic manifold, $J$ a possibly surface dependent complex structure, and $H$ a Hamiltonian on $M$. I am interested in a variant of Gromov-Floer convergence for solutions of Floer's equation:
$$(du - X_{H} \otimes \gamma) \circ j = J \circ (du - X_{H} \otimes \gamma) $$
for some subclosed one form $\gamma$ on our Riemann surface $(\Sigma,j)$. For simplicity let us suppose that $\Sigma$ is a cylinder with coordinates $f(s)$ and $\gamma=f(s)dt$.
The usual versions of Gromov Floer convergence that I have seen in say McDuff and Salamon's book prove that there is a $C^\infty$ limit (on the smooth part of the curves) given a family of solutions $u_q$ to Floer's equation for pairs $(H_q,J_q)$ which converge in the $C^\infty$ topology to a tuple $(H_\infty,J_\infty)$.
I have also come across this paper:
S. Ivashkovich and V. Shevchishin, Gromov compactness theorem for $J$-complex curves with boundary, Internat. Math. Res. Notices (2000), no. 22, 1167–1206
where they prove $C^0$ convergence to the homogeneous equation where $H=0$ provided we have a family of domain independent complex structures $J_q$ which converge in the $C^0$ topology to a complex structure $J_\infty$ (both in the closed case and the case of Lagrangian boundary). In the closed case, there is also an exposition of this by Tian and Siebert.
http://www.math.uni-hamburg.de/home/siebert/preprints/cime.pdf
Is there a version of this theorem (along with a reference) which allows us to incorporate surface-dependent complex structures and a family of Hamiltonians $H_q$ which converge in the $C^1$ topology to a limit $H_{\infty}$ ? I'd be equally interested in references with partial discussions e.g. that only discuss the case where no bubbles form or there is no breaking along Hamiltonian orbits.
