We develop automatic speech recognition (ASR) systems for stories told by Afrikaans and isiXhosa preschool children. Oral narratives provide a way to assess children's language development before they learn to read. We consider a range of prior child-speech ASR strategies to determine which is best suited to this unique setting. Using Whisper and only 5 minutes of transcribed in-domain child speech, we find that additional in-domain adult data (adult speech matching the story domain) provides the biggest improvement, especially when coupled with voice conversion. Semi-supervised learning also helps for both languages, while parameter-efficient fine-tuning helps on Afrikaans but not on isiXhosa (which is under-represented in the Whisper model). Few child-speech studies look at non-English data, and even fewer at the preschool ages of 4 and 5. Our work therefore represents a unique validation of a wide range of previous child-speech ASR strategies in an under-explored setting.