  We evaluate accurate low-field/low-temperature asymptotics of the thermal
conductivity perpendicular to magnetic field for one-band and two-band s-wave
superconductors using Keldysh-Usadel formalism. We show that heat transport in
this regime is limited by tunneling of quasiparticles between adjacent vortices
across a number of local points and therefore widely-used approximation of
averaging over circular unit cell is not valid. In the single-band case, we
obtain parameter-free analytical solution which provides theoretical lower
limit for heat transport in the mixed state. In the two-band case, we show that
heat transport is controlled by the ratio of gaps and diffusion constants in
different bands. Presence of a weaker second band strongly enhances the thermal
conductivity at low fields
