The following is a precise version of the "convergence of equations implies convergence of solutions" claim. (Even more general versions are available, but for parallel transport you just need autonomous equations.)

**Theorem**. Consider the family of ordinary differential equations indexed by $q\in Q$ (a topological space) given by 
$$ \dot{x}(t,q) = A(x(t),q) $$
where $x$ takes value in some $\Omega\subseteq \mathbb{R}^n$. Suppose we have:
- $A(x,q)$ is uniformly Lipschitz in $x$; more precisely there exists $K > 0$ such that for all $x_1, x_2\in \Omega$ and $q\in Q$ we have $\|A(x_1,q) - A(x_2,q)\| \leq K\|x_1 - x_2\|$. 
- $A(x,q)$ is uniformly bounded in $\Omega\times Q$.
- $A$ is equi-continuous in $q$; more precisely, for every $q_0\in q$ and $\epsilon > 0$ there exists a neighborhood $N$ of $q_0$ in $Q$ such that $\|A(x,q) - A(x,q_0)\| < \epsilon$ for every $x\in \Omega$. 

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 $t\in [0,1]$. 

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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). 

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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.