Elasticsearch: reindexing to replace null value

gistfile1.txt

#DELETE jmb-test
PUT jmb-test/transactions/1
{
  "user" : "jamie"
}
PUT jmb-test/transactions/2
{
  "user" : "kay"
}
PUT jmb-test/transactions/3
{
}

GET jmb-test/_search?q=!_exists_:user
# shows document 3

GET jmb-test/_search
{
  "aggs": {
    "site": {
      "terms": {
        "field": "user.keyword",
        "size": 10,
        "missing": "MISSING USER"
      }
    }
  }
}
# Shows a separate bucket for MISSING USER

PUT _template/jmb-test-reindex
{
  "index_patterns": ["jmb-test-reindex"],
  "mappings": {
    "transactions": {
      "properties": {
        "user": {
          "type": "text",
          "fields": {
            "keyword": {
              "type": "keyword",
              "null_value": "MISSING"
            }
          }
        }
      }
    }
  }
}

POST _reindex
{
  "source": {
    "index": "jmb-test"
  },
  "dest": {
    "index": "jmb-test-reindex"
  }
}

GET jmb-test-reindex
# Shows the expected mapping in place
GET jmb-test-reindex/_search?q=!_exists_:user
# still shows doc#3 as having the field missing
GET jmb-test-reindex/_search?q=user.keyword:MISSING
# 0 docs returned

GET jmb-test-reindex/_search
{
  "aggs": {
    "user": {
      "terms": {
        "field": "user.keyword",
        "size": 10,
        "missing": "MISSING USER"
      }
    }
  }
}
# Shows again a separate bucket for MISSING USER