We know about Kolmogorov Criterion for the tightness of a stochastic process $X_n(t)$

1.The sequence $(X_{n}(0))_{n\geq0}$ is tight.

2.There exist constants $\gamma\geq0$,$\alpha>1$, $K>0$ and an integer $n_0$ such that $$E(|X_{n}(t_{2})-X_{n}(t_{1})|^{\gamma})\leq K|t_{2}-t_{1}|^{\alpha}, \forall n \geq n_0$$ for all $t_{1},t_{2}$.

My first question: what should the $n_0$ depend? Could it depend on the $t_{1}$ and $t_{2}$?

My second question: Is there any other criterion for tightness with the parameter $\alpha=1$ for the version of the moment condition?