= Solution
Follow successive <photons> from the same comoving source. Their <comoving radial distance> is fixed, so
$$
\chi=\int_{t_e(t_0)}^{t_0}\frac{c\,dt}{a(t)},\qquad
0=\frac c{a(t_0)}-\frac c{a(t_e)}\frac{dt_e}{dt_0}.
$$
It follows that $dt_e/dt_0=a(t_e)/a(t_0)=1/(1+z)$. Taking the <logarithmic derivative> of $1+z=a(t_0)/a(t_e)$, with both endpoints allowed to vary, gives
$$
\frac{\dot z}{1+z}=H(t_0)-H(t_e)\frac{dt_e}{dt_0}.
$$
Therefore \b[the observer-time <redshift drift> is]
$$
\boxed{\dot z=H(t_0)(1+z)-H(t_e).}
$$
Holding the emission time fixed would omit the second term and would not compare successive observations of the same source.
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