I think you will need to use a script processor. None of the processors know how to deal with a nested field (which appears as an array of sub-maps). Eg, for doing lowercase with your example:
"processors" : [
{
"script" : {
"source": "for (Map nested : ctx._source['nest']) { nested['field_lower'] = nested['field_raw'].toLowerCase(Locale.ROOT); }"
}
}
]