Boundary string field theory is an open string field theory which has been originally formulated on a flat target space. In this talk I present recent progress in the study of BSFT in curved space backgrounds. Starting from a factorization property of the associated path-integral, non-local open string couplings can be identified which implement shifts in the closed string background. This generally affects the stability of D-branes, which is analyzed with renormalisation group methods. Evidence for the conjectured flow towards D-branes in curved space is presented for the example of a SU(2) WZW model.