How to correct observed flux densities for redshift
You are familiar with the relationship between flux and luminosity, right? This is usually taken as the definition of luminosity distance in extragalactic astronomy,
$$ F = \frac{L}{4\pi D_L^2}.$$
The quickest way to convert this into a relationship with spectral variables is to use differential 1-forms. So, $L \rightarrow L_\lambda \,\mathrm{d}\lambda_e$ and $F \rightarrow F_\lambda \,\mathrm{d}\lambda_o$, with $\lambda_e$ the photon wavelength at emission (also called the rest frame wavelength) and $\lambda_o$ the observed wavelength at $z=0$ (also called the comoving wavelength). Thus
\begin{align}
  F_\lambda \,\mathrm{d}\lambda_o &amp; = \frac{L_\lambda \,\mathrm{d}\lambda_e}{4\pi D_L^2} \Rightarrow\\
  F_\lambda &amp; = \frac{L_\lambda}{4\pi D_L^2} \times \frac{\mathrm{d}\lambda_e}{\mathrm{d}\lambda_o}. 
\end{align}
Now you just need to take your derivative using the definition of redshift, $\lambda_o = (1+z)\lambda_e$, and use some way of calculating the luminosity distance in your cosmology.

Formally, a lot of people like to use $\nu F_\nu = \lambda F_\lambda$ as though it were a total flux, but it's not. The derivation will work, but only because redshifting scales all frequencies/wavelengths by a single constant. $\nu F_\nu$ isn't a measure of the flux under any part of the spectrum, it's a spectral flux using a different coordinate system. Consider the following
\begin{align}
   F &amp; = F_\nu \,\mathrm{d}\nu \\
    &amp;= \nu F_\nu \frac{\mathrm{d}\nu}{\nu} \\
    &amp;= F_{\ln \nu} \,\mathrm{d}\ln \nu.
\end{align}
In other words, $\nu F_\nu$ is still a spectral flux. If $F_\nu$ is in $\mathrm{ergs}\,\mathrm{cm}^{-2}\,\mathrm{Hz}^{-1}$ then $\nu F_\nu$ will be in $\mathrm{ergs}\,\mathrm{cm}^{-2}\,(e\text{-fold})^{-1}$. The $e\text{-fold}$ is a pseudo-unit, like radians, dex, octave, decibel, or magnitudes. In principle it can be omitted, just like how angular frequencies can be in radians per second, or just $\mathrm{s}^{-1}$.

Fun math exercise: show that $\ln(10) \nu F_\nu$ has units of $\mathrm{ergs}\,\mathrm{cm}^{-2}\,\mathrm{dex}^{-1}$.