  Magnetic topological phases of quantum matter are an emerging frontier in
physics and material science. Along these lines, several kagome magnets have
appeared as the most promising platforms. Here, we explore magnetic
correlations in the transition-metal-based kagome magnet TbMn$_{6}$Sn$_{6}$
using muon spin rotation, combined with local field analysis and neutron
diffraction. Our results show that the system exhibits an out-of-plane
ferrimagnetic structure $P6/mm'm'$ (comprised by Tb and Mn moments) with slow
magnetic fluctuations below $T_{\rm C2}$~=~320~K. These fluctuations exhibit a
slowing down below $T_{\rm C1}^{*}$~${\simeq}$~120~K, and we see the formation
of static patches with ideal out-of-plane order below $T_{\rm
C1}$~${\simeq}$~20~K which grow in a volume with decreasing temperature. The
appearance of the static patches has a similar onset to the interesting
phenomenon such as spin-polarized Dirac dispersion with a large Chern gap and
topological edge states. We further show that the temperature evolution of the
anomalous Hall conductivity (AHC) is strongly influenced by the low temperature
magnetic crossover. Our presented experimental results show that the onset of
the topological electronic properties tied to the Dirac band is promoted only
by true static out-of-plane ferrimagnetic order in TbMn$_{6}$Sn$_{6}$ and is
washed out by the slow magnetic fluctuations above $T_{\rm
C1}$~${\simeq}$~20~K. Remarkably, hydrostatic pressure of 2.1 GPa stabilises
static out-of-plane topological ferrimagnetic ground state in the whole volume
of the sample. Therefore the exciting perspective arises of a magnetic system
in which the topological response can be controlled, and thus explored, over a
wide range of parameters.
