= Solution
Write $g_t=g_{K_t}$ and, for $s<t$, let $K_{s,t}$ be the filled image hull $g_s(K_t\setminus K_s)$. The <half-plane-capacity composition rule> gives
$$
g_t=g_{K_{s,t}}\circ g_s,\qquad\operatorname{hcap}(K_{s,t})=2(t-s).
$$
The <Loewner local growth property> says that these mapped increments shrink uniformly on bounded time intervals: for every $A<\infty$,
$$
\omega_A(h):=\sup_{0\leq s<t\leq A,\ t-s\leq h}\operatorname{rad}(K_{s,t})\longrightarrow0
\quad\text{as }h\downarrow0.
$$
Using <diameter> instead of radius is equivalent, since a nonempty <compact H-hull> has its closure meeting the real axis. This is a condition on the increments after mapping out the past, not merely on the Euclidean size of $K_t\setminus K_s$ in the original domain.
For fixed $t$, the nonempty compact sets $\overline{K_{t,t+h}}$, $h>0$, are nested as $h$ decreases. Their diameters tend to zero by <Loewner local growth property>. The nested-compact-set property gives a single intersection point, and its imaginary part is zero because $K_{t,t+h}$ lies in a real-centred disc of radius tending to zero. Define the <Loewner transform> by
$$
\boxed{\{\xi_t\}=\bigcap_{h>0}\overline{K_{t,t+h}},\qquad\xi_t\in\mathbb R.}
$$
One may restrict the intersection to sufficiently small $h$. Every point of $K_{t,t+h}$ is within $2\omega_A(h)$ of $\xi_t$ when $t+h\leq A$.
Here is a direct proof of <continuity>, using the permitted displacement estimate. Fix $s<t$ and put $\delta=t-s$. Take $0<h\leq\delta$ and a point $v\in K_{t,t+h}$. Set $z=g_{K_{s,t}}^{-1}(v)$. The hull composition rule puts $z\in K_{s,t+h}$, while the displacement estimate gives
$$
|v-z|\leq C\operatorname{rad}(K_{s,t})\leq C\omega_A(\delta).
$$
Since $z$ is within $2\omega_A(\delta+h)$ of $\xi_s$, let $h\downarrow0$, taking $v\to\xi_t$, to obtain
$$
\boxed{|\xi_t-\xi_s|\leq C\omega_A(\delta)+2\omega_A(2\delta).}
$$
Use a slightly larger fixed time interval if $t$ is its endpoint. Both terms tend to zero with $\delta$. This proves <continuity> from both sides, and indeed <uniform continuity> on every compact time interval. The real <boundary> value is obtained as a limit; evaluating the displacement estimate directly on a possibly rough <boundary> is unnecessary.
The <Loewner correspondence theorem> states that a capacity-$2t$ increasing hull family with <Loewner local growth property> has the unique continuous real <Loewner driving function> just defined, and its <mapping-out functions> solve the <Chordal Loewner equation>
$$
\boxed{\partial_tg_t(z)=\frac2{g_t(z)-\xi_t},\qquad g_0(z)=z.}
$$
For each $z\in\mathbb H$, this solution is taken up to its maximal lifetime; $\mathbb H\setminus K_t$ is precisely the set of points whose lifetimes exceed $t$. Conversely a continuous real driver gives such a hull family through this <ordinary differential equation>. A continuous driver alone need not have a continuous <Loewner trace>; that additional assertion is part of the trace theory of <SLE>, not of the deterministic correspondence theorem.
A continuous process $\gamma:[0,\infty)\to\overline{\mathbb H}$ is chordal <Schramm–Loewner evolution> with parameter $\kappa$ from $0$ to infinity when $\gamma_0=0$, its filled past generates these capacity-$2t$ hulls, and their <Loewner driving function> is
$$
\boxed{\xi_t=\sqrt\kappa\,B_t,}
$$
with $B$ standard real <Brownian motion>. The filling includes bounded components cut off from infinity. Equivalently its <Loewner trace> is the continuous extension of $\gamma_t=\lim_{y\downarrow0}g_t^{-1}(\xi_t+iy)$. This last <boundary> limit is the trace-existence input for <SLE>.
Fix $r>0$ and scale space and time together:
$$
\widehat K_t=r^{-1}K_{r^2t},\qquad
\widehat g_t(z)=r^{-1}g_{r^2t}(rz),\qquad
\widehat\xi_t=r^{-1}\xi_{r^2t}.
$$
The <half-plane capacity> of $\widehat K_t$ is $2t$ by part (a). Differentiation gives
$$
\partial_t\widehat g_t(z)=\frac2{\widehat g_t(z)-\widehat\xi_t}.
$$
By <Brownian scaling>, $r^{-1}B_{r^2t}$ is again standard <Brownian motion>, so the new driver has the same law as the old one. Uniqueness of the <Chordal Loewner equation> transfers this equality to the hulls and their continuous <Loewner traces>. Thus
$$
\boxed{(r^{-1}\gamma_{r^2t})_{t\geq0}\ \overset{d}=\ (\gamma_t)_{t\geq0}.}
$$
This is <capacity-parametrized scale invariance of a Loewner chain>. It includes $\kappa=0$, when the driver is zero and the trace is the deterministic vertical slit $\gamma_t=2i\sqrt t$.
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