Move data from one array to another array in the same document

Here is a working sample for Mongo version 4.4, our strategy will be to first match the employment we want to move, then update each of the persons accordingly, I will say that doing this in Mongo v5.0+ will be much cleaner as we could use $getField, this would make the first two stages of the pipelined update much simpler.

With that said here is the code sample:

const updatePayload = {
   fromPerson: "61cd90594947be000838f7c1",
   toPerson: "61cd90594947be000838f7c2",
   employmentId: "61cd9059494abe000838f7c8",
}

db.collection.update({},
[
  {
    $set: {
      tmpPerson: {
        $arrayElemAt: [
          {
            $filter: {
              input: "$persons",
              as: "person",
              cond: {
                $eq: [
                  "$$person.personId",
                  updatePayload.fromPerson
                ]
              }
            }
          },
          0
        ]
      }
    }
  },
  {
    $set: {
      tmpEmployment: {
        $arrayElemAt: [
          {
            $filter: {
              input: "$tmpPerson.employment",
              as: "employment",
              cond: {
                $eq: [
                  "$$employment.employmentId",
                  updatePayload.employmentId
                ]
              }
            }
          },
          0
        ]
      }
    }
  },
  {
    $set: {
      persons: {
        $map: {
          input: "$persons",
          as: "person",
          in: {
            $switch: {
              branches: [
                {
                  case: {
                    $eq: [
                      "$$person.personId",
                      updatePayload.fromPerson
                    ]
                  },
                  then: {
                    $mergeObjects: [
                      "$$person",
                      {
                        employment: {
                          $filter: {
                            input: "$$person.employment",
                            as: "employment",
                            cond: {
                              $eq: [
                                "$$employment.employmentId",
                                updatePayload.employmentId
                              ]
                            }
                          }
                        }
                      }
                    ]
                  }
                },
                {
                  case: {
                    $eq: [
                      "$$person.personId",
                      updatePayload.toPerson
                    ]
                  },
                  then: {
                    $mergeObjects: [
                      "$$person",
                      {
                        employment: {
                          $concatArrays: [
                            "$$person.employment",
                            [
                              "$tmpEmployment"
                            ]
                          ]
                        }
                      }
                    ]
                  }
                }
              ],
              default: "$$person"
            }
          }
        }
      }
    }
  },
  {
    $unset: [
      "tmpPerson",
      "tmpEmployment"
    ]
  }
])

Mongo Playground

One last thing to note is that this pipeline assumes that the input exists properly, otherwise it will add null to the employment array. if this is not the case then you should add some protection.