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High resolution heat capacity and thermal expansion experiments performed onsingle crystalline kagome-staircase compounds (Co$_{1-x}$Ni$_x$)$_3$V$_2$O$_8$are presented. The parent compounds Co$_3$V$_2$O$_8$ and Ni$_3$V$_2$O$_8$undergo a complex sequence of first- and second-order magnetic phasetransitions. The low-temperature ($T<40$ K) magnetic entropy evolvesmonotonously with the doping content $x$, from the full S=1 Ni$^{2+}$ magneticentropy in Ni$_3$V$_2$O$_8$ to half of the S=3/2 Co$^{2+}$ magnetic entropy inCo$_3$V$_2$O$_8$. Thermal expansion coefficients $\alpha_i$ ($i = a$, $b$ and$c$) show a strong anisotropy for all (Co$_{1-x}$Ni$_x$)$_3$V$_2$O$_8$compounds. The low-temperature magnetic distortion indicates that Co-doping(Ni-doping) has similar effects to applying a uniaxial pressures along $a$ or$b$ ($c$). Linear Gr\"{u}neisen parameters $\Gamma_i$ are extracted for thethree main axes $i$ and exhibit a complex behavior with both temperature anddoping. For each axis, $\Gamma_i$ and $\alpha_i$ exhibit a sign change (at lowtemperature) at the critical concentration $x_c\simeq0.25$, at which theincommensurate magnetic propagation vector changes. Beyond our study, anunderstanding of the multiple and complex parameters (magnetic frustration,magnetic anisotropy, mixture of S=1 and S=3/2 ions, etc.) is now necessarily tobring light to the rich magneto-elastic properties of(Co$_{1-x}$Ni$_x$)$_3$V$_2$O$_8$.

Complex magneto-elastic properties in the frustrated kagome-staircase compounds (Co$_{1-x}$Ni$_x$)$_3$V$_2$O$_8$