動画

単にLineの頂点を移動するだけでは移動速度が頂点位置のばらつきに依存してしまうので、Line上を等速で滑らかに移動できるようにします。
LineRenderer上を動かしたいオブジェクトに下記のスクリプトをアタッチすると、オブジェクトがライン上を移動します。

using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class MoveOnLine : MonoBehaviour
{
    [SerializeField] LineRenderer m_line=null;
    [SerializeField,Tooltip("速度[m/sec]")] float m_speed = 0.1f;
    [SerializeField,Tooltip("ライン上を動くObject")] Transform m_targetTr = null;
    int m_linePtr;
    float[] m_costArr;
    float m_remain;

    // Start is called before the first frame update
    void Start()
    {
        if (m_targetTr == null)
        {
            m_targetTr = transform;
        }
        init();
    }

    // Update is called once per frame
    void Update()
    {
        float delta = m_speed * Time.deltaTime;
        while (delta > 0f)
        {
            if (m_remain > delta)
            {
                m_remain -= delta;
                delta = 0f;
                break;
            }
            else
            {
                delta -= m_remain;
                m_linePtr = (m_linePtr + 1) % m_line.positionCount;
                m_remain = m_costArr[m_linePtr];
            }
        }
        Vector3 basePos = m_line.GetPosition(m_linePtr);
        if (m_remain > 0f)
        {
            float rate = 1f- m_remain / m_costArr[m_linePtr];
            basePos += (m_line.GetPosition((m_linePtr + 1) % m_line.positionCount) - basePos) * rate;
        }
        if (!m_line.useWorldSpace)
        {
            basePos = m_line.transform.position + Vector3.Scale(m_line.transform.rotation * basePos, m_line.transform.lossyScale);
        }
        m_targetTr.position = basePos;
    }

    private void init()
    {
        m_costArr = new float[m_line.positionCount];
        Vector3 pos = m_line.GetPosition(0);
        for (int i = 0; i < m_line.positionCount; ++i)
        {
            Vector3 nextPos = m_line.GetPosition((i + 1)% m_line.positionCount);
            m_costArr[i] = (nextPos - pos).magnitude;
            pos = nextPos;
        }
        if (!m_line.loop)
        {
            m_costArr[m_line.positionCount - 1] = 0f;
        }
        m_linePtr = 0;
        m_remain = m_costArr[0];
    }
}