There used to be one with a change in Daugavpils But iam not sure if they still operate please check the Website of the Railcompanies of these Countries ;) 

By BUS yes, currently by train NOt-that one via Dgvpls was weekends only.

Best site for all this: seat61.com , as its so much pro-train it goes to pains in finding ways to use them.

These new countries are so thinly populated and have so few demand for such travels, that the offer is minimal to nil. BUSes are far better suited for that. In the distant past the major lines all went to the capital of the then existing world; MOCKBA. That is no more.

If wanting to go onward: Riga-Tallinn can as such be done, if one insists, by 2 trains-change at border point=Valga-the wait there is even longer as the direct bus needs!