The following is a precise version of the "convergence of equations implies convergence of solutions" claim.
Theorem. Let $\Omega\subseteq \mathbb{R}^n$ be open, and $Q\subseteq \mathbb{R}$. Let $A:\Omega\times [0,1]\times Q\to \mathbb{R}^n$ be continuous. Consider the family of ordinary differential equations indexed by $q\in Q \subseteq \mathbb{R}$ given by $$ \dot{x}(t,q) = A(x(t),t,q) $$ Suppose we have:
- $A(x,t,q)$ is uniformly Lipschitz in $x$; more precisely there exists $K > 0$ such that for all $x_1, x_2\in \Omega$ and $q\in Q$ and $t\in [0,1]$ we have $\|A(x_1,t,q) - A(x_2,t,q)\| \leq K\|x_1 - x_2\|$.
- $A(x,t,q)$ is uniformly bounded in $\Omega\times Q$.
Let $q_k$ be a sequence converging to $q_\infty$ in $Q$, and suppose for each $k$, we have a solution $x(\cdot ,q_k):[0,1]\to \Omega$ to the ODE above. If furthermore $x(0,q_k)$ converges to $x(0,q_\infty)$, then $x(t,q_k)$ converges to $x(t,q_\infty)$ for every $t\in [0,1]$.
In the literature such results are usually called "continuous dependence on parameters" for ordinary differential equations. Henri Cartan devoted quite a few pages in his book Differential Calculus to this (I don't have my copy with me to track down a precise theorem number).
For parallel transport, the first condition is trivial since the relevant operator is linear, the second is also easy to check once you localize to a small neighborhood of the curve $\gamma$ and locally trivialize. The third condition is the only one that requires the curves to converge in $C^1$, since in local coordinates the connection coefficients depend continuously on the position of the curve and is linear on the velocity of the curve.