Paul Kirk defined in the late eighties a link-homotopy invariant of 2-component link maps, which are 'self-immersed spheres' in 4-space. Recently, Schneiderman and Teichner showed that this 'Kirk invariant' is in fact a complete link-homotopy invariant. In this talk, I will explain how to extend Kirk's construction to link maps of any number of components, in an explicit and computable way. This is a joint work with B. Audoux and A. Yasuhara.