Mark, Caritas and HEKS would be a good place to start because apparently every asylum-seeker has the right to be accompanied by a representative of an official asylum-advocacy organization (eidgenoessisch anerkanntes Fluchtingshilfeswerk - hope I've done the expression justice in my translation) during any official hearings or dealings with the authorities, and Caritas and HEKS are very active in this. A friend of Mr. B's did this for awhile, and said it was eye-opening how badly some of these people are treated...
Asylum-seekers and displaced families are a big focus area for Zurich's Red Cross chapter, for instance the Big Brother/Big Sister program is aimed at children of asylum-seekers.
Perhaps it's worth checking with other, local faith-based organizations besides Caritas and HEKS, who have made working with asylum-seekers a priority?