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We consider a class of Kirchhoff type reaction-diffusion equations with variable exponents and source terms $$u_t-M\left({\int }_{\Omega }{\left|\nabla u\right|}^2\mathrm{d}x\right)\Delta u+{\left|u\right|}^{m\left(x\right)-2}{u}_t={\left|u\right|}^{r\left(x\right)-2}u.$$ We prove with suitable assumptions on the variable exponents $r(·),$ $m(·)$ the global existence of the solution and a stability result using potential and Nihari’s functionals with small positive initial energy, the stability being based on Komornik’s inequality.