Need to serve; existing case; respondent out of country; self in state other than where case originated. How?

I'm not familiar with Oregon procedure, but in the jurisdictions in which I practice there is no need for the formalities of service of process on a party to a case which has already been commenced.  Once a case has started. service on a party of papers in that case is accomplished by merely filing the document to be served with the court's efiling system or by mail upon that party's attorney (if they are not represented by an attorney, than mail it to the party themselves).  If you do need to serve the papers they way you would serve process (e.g. a summons), you would have to serve them pursuant to whatever treaty governs service or process in tha country (most countries are party to the Hague Convention treaty relating to service or process, Canada being a notable exception).  You should find out whether you need to serve in the way you serve process by asking the clerk of the court.  If you do need to, you may be able to find out what you need to know by calling the embassy for the country in which your former spouse now resides.