= Solution
Let $\mathbf x'=\mathbf x-\mathbf vt$, $t'=t$ for a constant relative <velocity> $\mathbf v$. The <Galilean transformation> gives $\nabla'=\nabla$ and $\partial_{t'}=\partial_t+\mathbf v\cdot\nabla$. In the <magnetic Galilean limit of the pre-Maxwell equations>, take
$$
\boxed{\mathbf B'=\mathbf B,\qquad \mathbf J'=\mathbf J,\qquad \mathbf E'=\mathbf E+\mathbf v\times\mathbf B.}
$$
The <Ampère's law> without <displacement current> and the solenoidal constraint are immediately unchanged. For <Faraday's law>, the constant-<velocity> vector identity is $\nabla\times(\mathbf v\times\mathbf B)=\mathbf v\nabla\cdot\mathbf B-(\mathbf v\cdot\nabla)\mathbf B$. Thus
$$
-\nabla'\times\mathbf E'=-\nabla\times\mathbf E+(\mathbf v\cdot\nabla)\mathbf B=\partial_{t'}\mathbf B'.
$$
These are transformations of the magnetic nonrelativistic approximation. The exact charge-current transformation also contains the convective term $-\rho_e\mathbf v$ in $\mathbf J'$; ignoring it requires the quasineutral magnetic limit. They are not exact transformations of the full <Maxwell equations>.
Since $\mathbf u'=\mathbf u-\mathbf v$, the combination $\mathbf E'+\mathbf u'\times\mathbf B'$ equals $\mathbf E+\mathbf u\times\mathbf B$. For an ideally conducting fluid the rest-frame <electric field> vanishes, so the ideal <moving-conductor Ohm law> is $\mathbf E+\mathbf u\times\mathbf B=0$. Substitution in <Faraday's law> gives
$$
\boxed{\partial_t\mathbf B=\nabla\times(\mathbf u\times\mathbf B),\qquad\nabla\cdot\mathbf B=0.}
$$
The <ideal magnetohydrodynamic induction equation> requires nonrelativistic fluid motion, negligible <displacement current>, and negligible nonideal terms in the electric-field relation. In particular, <magnetic diffusion> must be slow compared with advection: the <magnetic Reynolds number> $\mathrm{Rm}=UL/\eta_m\gg1$, where $\eta_m=(\mu_0\sigma)^{-1}$ is the <magnetic diffusivity>. The single-fluid <moving-conductor Ohm law> must apply on the scales considered, with corrections to that relation negligible; viscosity is not itself excluded by this induction equation.
For a smooth ideal evolution, <magnetic flux freezing> states that the <magnetic flux> through every material surface is conserved and that <magnetic field lines> are carried with the fluid. Their connectivity is preserved: ideal smooth advection does not reconnect them. These implications are stated here without proof, as requested.
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