AFT models in Stata with time-varying covariates The AFT models from Stata's streg can handle time-varying covaraites. In particular, the manual states that

streg is suitable only for data that have been stset. By stsetting your data, you define the
variables _t0, _t, and _d, which serve as the trivariate response variable $(t_0, t, d)$. Each response
corresponds to a period under observation, $(t_0, t]$, resulting in either failure $(d = 1)$ or right-censoring
$(d = 0)$ at time $t$. As a result, streg is appropriate for data exhibiting delayed entry, gaps, time-varying
covariates, and even multiple-failure data.

The question is how this actually works? Suppose that I use a log-normal distribution. Then the density and survival function without time-varying covariates is
$$
\begin{align*}
f(t\mid \vec x)&= \frac 1{t\sigma\sqrt{2\pi}}\exp\left(-
  \frac{(\log t - \vec x^\top\vec\beta)^2}{2\sigma^2}\right) \\
S(t \mid \vec x) &= 1 - \Phi\left(
     \frac{\log t - \vec x^\top\vec\beta}{\sigma}
  \right)
\end{align*}
$$
and the hazard is given by $\lambda (t\mid\vec x) = f(t\mid\vec x)S(t \mid \vec x)^{-1}$.
This is fairly easy to interpret as it states that $\log T \mid \vec x$ is normally distributed with mean $\vec x^\top\vec\beta$ and variance $\sigma^2$.
What I presume that what streg does is the following: Suppose that $\vec x(t)$ is the time-varying covariates which is modeled as having jumps as is typical. Then the
assumptions is the the hazards is given by $\lambda (t\mid\vec x(t)) = f(t\mid\vec x(t))S(t \mid \vec x(t))^{-1}$. Is this correct? If not, how does streg handle the time-varying covariates?

Given EdM's answer, the model is in counting process setup like I state above. Thus, suppose that we look at left-truncation time $t_i$ with covariate $\vec x_i$. Let
$$
D(t, \vec x) = \Phi\left(
  \frac{\log t - \vec x^\top\vec\beta}{\sigma}
  \right)
$$
Then the density, CDF, and survival function are:
$$
\begin{align*}
  f(t\mid\vec x_i, t_i) &= \begin{cases}
    \frac{f(t\mid \vec x_i)}{S(t_i \mid \vec x_i)} & t \geq t_i \\
    0 & t < t_i
  \end{cases} \\
  F(t \mid \vec x_i, t_i) &= \begin{cases}
   \frac{D(t, \vec x_i) - D(t_i, \vec x_i)}{S(t_i \mid \vec x_i)} 
     & t \geq t_i \\
    0 & t < t_i
  \end{cases} \\
&= \begin{cases}
   \frac{D(t, \vec x_i) - D(t_i, \vec x_i)}{1 - D(t_i, \vec x_i)} 
     & t \geq t_i \\
    0 & t < t_i
  \end{cases} \\
S(t\mid\vec x_i, t_i) &= 1 - F(t \mid \vec x_i, t_i) =
  \begin{cases}
   \frac{1 - D(t, \vec x_i)}{1 - D(t_i, \vec x_i)} 
     & t \geq t_i \\
    1 & t < t_i
  \end{cases} \\ 
&= \begin{cases}
   \frac{S(t, \vec x_i)}{S(t_i, \vec x_i)} 
     & t \geq t_i \\
    1 & t < t_i
  \end{cases}
\end{align*}
$$
Thus, the hazard is
$$
\lambda(t\mid \vec x_i, t_i) = 
  \frac{f(t\mid\vec x_i, t_i)}{S(t\mid\vec x_i, t_i)} 
  = \frac{f(t\mid\vec x_i)}{S(t\mid\vec x_i)} 
  = \lambda(t \mid \vec x_i)
$$
as I stated in my question.
The distribution of the survival time is not that obvious I guess but I figure that it is best expressed in terms of the hazards. In particular, suppose that covariates are given by:
$$
\vec x(t) = \begin{cases}
  \vec x_1 & t < t_1 \\ 
  \vec x_2 & t_1 \leq t < t_2 \\
  \vdots & \vdots \\
  \vec x_k & t_{k - 1} \leq t < t_k
\end{cases} 
$$
for some $k \geq 1$. Then the hazard is
$$
\lambda(t \mid \vec x(t)) = \begin{cases}
  \lambda(t\mid \vec x_1) & t < t_1 \\ 
  \lambda(t\mid \vec x_2) & t_1 \leq t < t_2 \\
  \vdots & \vdots \\
  \lambda(t\mid \vec x_k) & t_{k - 1} \leq t < t_k
\end{cases} 
$$

In the data format you describe (also called "counting process" format), an observation is considered left-truncated at $t_0$ (no information until after that time).
Your formula is the definition of the instantaneous hazard, given covariate values. Technically, the software to solve a parametric AFT model finds parameter values that maximize the likelihood of the data (covariate values and event times).
For a case left-truncated at $t_0$ having an event at $t$, the contribution to likelihood is proportional to $f(t)/S(t_0)$, with $f(t)$ and $S(t_0)$ determined for the corresponding covariate values.* That's the classic parametric modeling approach (which I assume Stata implements).

*This page summarizes the likelihood expressions for combinations of events and left or right censorings and truncations.
