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 $t\in [0,1]$.

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.

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.