The C# script makes the camera move on the path desired in Unity3D.

AutoCamera.cs

/* 
 * author : jiankaiwang
 * description : The C# script makes the camera move on the path desired in Unity3D.
 * platform : Unity
 * date : 2018/01
 */

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

// struct for controling camera moving and rotating
public struct CamNavCtlPoint {
    public Vector3 Position;
    public Quaternion Rotation;
}

// struct recording the start and end indexes within the position
public struct PosArray {
    public int start, end;
}

// moving unit for each dimensions
public struct PosIncreament {
    public float X, Y, Z;
}

public class AutoCamera : MonoBehaviour {

    // the step unit between two positions
    public int step = 100;
    public int initX = 0, initY = 0, initZ = 0;

    // use stepIndex to check whether the step is reached
    private int stepIndex = 1;
    private float moveX, moveY, moveZ;
    private int angleX, angleY, angleZ;
    private List<CamNavCtlPoint> ControlPoints = new List<CamNavCtlPoint>();
    private PosArray posIndex = new PosArray();
    private PosIncreament posIncreament = new PosIncreament();

    // Use this for initialization
    void Start() {
        // defined location
        ControlPoints.Add(new CamNavCtlPoint() { Position = new Vector3(initX, initY, initZ), Rotation = Quaternion.Euler(Vector3.zero) });
        ControlPoints.Add(new CamNavCtlPoint() { Position = new Vector3(10, 22, 10), Rotation = Quaternion.Euler(Vector3.zero) });
        ControlPoints.Add(new CamNavCtlPoint() { Position = new Vector3(20, 2, 10), Rotation = Quaternion.Euler(Vector3.zero) });

        // initialization
        moveX = 0;
        moveY = 0;
        moveZ = 0;
        posIndex.start = 0;
        posIndex.end = 1;
        posIncreament.X = 0.0f;
        posIncreament.Y = 0.0f;
        posIncreament.Z = 0.0f;
        angleX = 0;
        angleY = 0;
        angleZ = 0;
        calculateIncreamentUnit();
    }

    // get the increament amount for each dimensions on moving the position to the next one
    void calculateIncreamentUnit() {
        int startPos = posIndex.start;
        int endPos = posIndex.end;
        posIncreament.X = (ControlPoints[endPos].Position[0] - ControlPoints[startPos].Position[0]) / (float)step;
        posIncreament.Y = (ControlPoints[endPos].Position[1] - ControlPoints[startPos].Position[1]) / (float)step;
        posIncreament.Z = (ControlPoints[endPos].Position[2] - ControlPoints[startPos].Position[2]) / (float)step;
    }

    // get the new pair of start and end position index
    void indexMove() {
        posIndex.start = (posIndex.start + 1) % ControlPoints.Count;
        posIndex.end = (posIndex.start + 1) % ControlPoints.Count;

        // re-calculate the step unit
        calculateIncreamentUnit();
    }

    // Update is called once per frame
	void Update () {
        if (stepIndex++ > step) {
            indexMove();
            stepIndex = 1;
        }

        // --------------------------------------------------------------------
        // auto-move
        // --------------------------------------------------------------------
        // X : straffe += speed * Time.deltaTime;
        moveX = posIncreament.X;
        // Y
        moveY = posIncreament.Y;
        // Z : translation += speed * Time.deltaTime;
        moveZ = posIncreament.Z;
        transform.Translate(moveX, moveY, moveZ);

        // --------------------------------------------------------------------
        // auto-rotate
        // localRotation would execute the final code
        // --------------------------------------------------------------------
        // rotate on x-axis
        //angleX = (angleX + 1) % 360;
        //transform.localRotation = Quaternion.AngleAxis(angleX, Vector3.right);
        // rotate on y-axis
        //angleY = (angleY + 1) % 360;
        //transform.localRotation = Quaternion.AngleAxis(angleY, Vector3.up);
        // rotate on z-axis
        //angleZ = (angleZ + 1) % 360;
        //transform.localRotation = Quaternion.AngleAxis(angleZ, Vector3.forward);
    }
}