  The magnetism in the saw-tooth lattice of Mn in the olivine chalcogenides,
Mn$_2$SiS$_{4-x}$Se$_x$ ($x$ = 1$\textendash$4) is studied in detail by
analyzing their magnetization, specific heat and thermal conductivity
properties and complemented with density functional theory calculations. The
air-stable chalcogenides are antiferromagnets and show a linear trend in the
transition temperature, $T_N$ as a function of Se-content ($x$) which shows a
decrease from $T_N \approx$ 86~K for {\mss} to 66~K for {\msse}. Additional new
magnetic anomalies are revealed at low temperatures for all the compositions.
Magnetization irreversibilities are also observed as a function of $x$. The
specific heat and the magnetic entropy indicate the presence of short-range
spin fluctuations in Mn$_2$SiS$_{4-x}$Se$_x$. A spin-flop antiferromagnetic
phase transition in the presence of applied magnetic field is present in
Mn$_2$SiS$_{4-x}$Se$_x$, where the critical field for the spin flop increases
from $x$ = 0 towards 4 in a non-linear fashion. Density functional theory
calculations show that an overall antiferromagnetic structure with
ferromagnetic coupling of the spins in the $ab$-plane minimizes the total
energy. The band structures calculated for \mss\ and \msse\ reveal features
near the band edges similar to those reported for Fe-based olivines suggested
as thermoelectrics; however the experimentally determined thermal transport
data do not support superior thermoelectric features. The transition from
long-range magnetic order in \mss\ to short-range order and spin fluctuations
in \msse\ is explained using the variation of the Mn-Mn distances in the
triangle units that constitutes the saw-tooth lattice upon progressive
replacement of sulphur with selenium.
