Even for kāmāvacarakusala it is stated as:
Tattha katamo adhimokkhapaccayā bhavo? Ṭhapetvā adhimokkhaṁ, vedanākkhandho saññākkhandho saṅkhārakkhandho viññāṇakkhandho—ayaṁ vuccati “adhimokkhapaccayā bhavo”
Same as lokuttara kusala: no kammabhava, just registry in mind.
All punna kamma operate the akusala mula PS and generate rebirth. When removing defilements the kusala mula PS operates (starting from sotapanna anugami?). Can it be that lokuttara kusala means that path factors are fulfilled and magga phala is reached? There it is mentioned:
Tattha katamo alobho …pe… adoso …pe… amoho? Yā paññā pajānanā …pe… amoho dhammavicayo sammādiṭṭhi dhammavicayasambojjhaṅgo maggaṅgaṁ maggapariyāpannaṁ