The following is an example: take a two state Markov chain with jump rates $\lambda_{0,1}=1$ and $\lambda_{1,0}=0$, and start it at state $0$. The initial variance is $0$, then increases, but as $t\to\infty$ the variance decreases.

The following is an example: take a Markov chain with states $0,\pm 1,\pm 1/2,\pm 1/3,..$ with jump rates $\lambda_{0,1}=\lambda_{0,-1}=\lambda_{1/k,1/(k+1)}=\lambda_{-1/k,-1/(k+1)}=1$ and rates $0$ otherwise, and start it at state $0$. The initial variance is $0$, then increases, but as $t\to\infty$ the variance decreases.