It should be independent of $t_1$ and $t_2$, of course. If it's not, the processes might be even discontinuous. For instance, $X_n$ is a Poisson process with parameter $1/n$. Then $E(|X_n(t_1)-X_n(t_2)|^2)\le |t_1-t_2|^2$ for all $n>|t_1-t_2|^{-1}$ (for all $n\ge 1$ if $t_1=t_2$).
And the same answer works for the second question: when $\alpha=1$, the processes need not to be continuous. In some special cases, where you have higher moments controlled by a lower polynomially, it may help (e.g. in the Gaussian case $\gamma=2$ and $\alpha=1$ is enough).