actionCreator.js

export const updateAlert = (alert, message) => {
  return {
    type: 'UPDATE_ALERT',
    alert,
    message
  }
}

export const updateAlertThenDelete = (alert, message) => {
  return function (dispatch) {
    return dispatch(
      updateAlert(alert, message).then(() => {
        console.log('done');
      })
    )
  }
}

component.js

this.props.updateAlertThenDelete('error', ERROR_MESSAGE)

@ddsol, you're right. I think I was overthinking this. I don't even need a callback in this instance.. all I need to do is:

  waitTurn() {
    const ERROR_MESSAGE = 'Please wait your turn.';

    this.props.updateAlert('error', ERROR_MESSAGE)
    setTimeout(() => {
      console.log('remove alert code here');
    }, 5000)
  }

Thank you for explaining everything to me, it's now starting to make sense. I appreciate it!