  We study the semiclassical kinetics of 2D fermions in a smoothly varying
magnetic field $B({\bf r})$. The nature of the transport depends crucially on
both the strength $B_0$ of the random component of $B({\bf r})$ and its mean
value $\bar{B}$. For $\bar{B}=0$, the governing parameter is $\alpha=d/R_0$,
where $d$ is the correlation length of disorder and $R_0$ is the Larmor radius
in the field $B_0$. While for $\alpha\ll 1$ the Drude theory applies, at
$\alpha\gg 1$ most particles drift adiabatically along closed contours and are
localized in the adiabatic approximation. The conductivity is then determined
by a special class of trajectories, the "snake states", which percolate by
scattering at the saddle points of $B({\bf r})$ where the adiabaticity of their
motion breaks down. The external field also suppresses the diffusion by
creating a percolation network of drifting cyclotron orbits. This kind of
percolation is due only to a weak violation of the adiabaticity of the
cyclotron rotation, yielding an exponential drop of the conductivity at large
$\bar{B}$. In the regime $\alpha\gg 1$ the crossover between the snake-state
percolation and the percolation of the drift orbits with increasing $\bar{B}$
has the character of a phase transition (localization of snake states) smeared
exponentially weakly by non-adiabatic effects. The ac conductivity also
reflects the dynamical properties of particles moving on the fractal
percolation network. In particular, it has a sharp kink at zero frequency and
falls off exponentially at higher frequencies. We also discuss the nature of
the quantum magnetooscillations. Detailed numerical studies confirm the
analytical findings. The shape of the magnetoresistivity at $\alpha\sim 1$ is
in good agreement with experimental data in the FQHE regime near $\nu=1/2$.
