The problem with Dürst is that it is a fairly common name in canton Glarus - with citizenships since before 1800 in
- Braunwald
- Diesbach
- Ennenda
- Glarus
- Linthal
- Mitlödi
- Mühlehorn
- Obstalden
- Riedern
- Sool
The best source to search for your ancestor would be
Das Genealogienwerk von Johann Jakob Kubly-Müller (scroll down) - a genealogical collection covering the entire canton. This source (manuscript) is only available at the State Archives, however, with some restrictions on more recent data. To my knowledge the Archive does not do searches in this source - you may have to find someone in the area to do this for you.
Church records are not really an alternative for researchers overseas - to my knowledge have not been filmed.
Someone emigrating after 1876 should be registered with the Civil Registry Office (Zivilstandsamt), however: try to contact
this office and ask them if they can identify your ancestor - and (if yes) what they expect you to do to get a
Familienschein.