If $X_1, X_2, \dots$ are i.i.d.random variables with a continuous distribution function $F$ and empirical distribution function $F_n$ we can construct a process which converges in distribution to a Brownian Bridge (by Donsker's theorem):
$$G_n(x) = \sqrt n (F_n(x) - F(n))\text{.}$$
Now, my problem is that I don't have values from a fixed distribution $F$. Instead there are distributions $H_n$ which converge in distribution to $F$. From each of those $H_n$ I have $n$ values. Again, $H_{n,n}$ is the empirical distribution function and I can define a process:
$$G_n = \sqrt n (H_{n,n}(x) - F(x))\text{.}$$
Do the $G_n$ still converge to brownian bridges? And if not in general, are there additional properties which make this converge? Pitfalls?