SOURCE

var vm = new Vue({
  el: '#app',
  data: function(){
    return {
      arr: [{arg: 'arr1', id: 0, key: 0}, {arg: 'arr2', id: 1, key: 1}, {arg: 'arr3', id: 2, key: 2}]
    }
  },
  methods: {
    changeArr: function(){
      var arr = this.arr
      var arg =  arr.shift()
      // arg.id += 3 
      arr.push(arg)
      this.arr = arr
    }
  }
})

<div id="app">
  <ul>
    <li v-for="(item,index) of arr" :id="item.id" :key="item.key">{{item.arg}}<input type="text" /></li>
    <button @click="changeArr">click</button>
  </ul>
</div>

#app{
  position: absolute;
  margin: auto;
  top: 0;
  left: 0;
  right: 0;
  max-width: 750px;
  text-align: center;
}