a script for orthogonal cameras, which starts moving the camera when the player is about to leave the visible region. can be seen here: https://gfycat.com/NecessaryFelineHare

SomeCameraTest.cs

using UnityEngine;

public class SomeCameraTest : MonoBehaviour {
    private Bounds outer_bounds;
    private Bounds inner_bounds;
    private bool bounds_reached = false;

    [SerializeField] private Transform target;
    [SerializeField] private Camera cam;
    [SerializeField] private float camSpeed = 8f; // set to player speed
    [SerializeField] private Vector2 innerBoundsOffset = new Vector2(3, 3);

    private Vector3 velocity;

    void Start() {
        outer_bounds = new Bounds(new Vector3(0, 0, 0), new Vector3(2*cam.orthographicSize * cam.aspect, 10, 2*cam.orthographicSize));
        outer_bounds.Expand(new Vector3(-1, 0, -1)); // seems to be nicer, maybe even -2.

        inner_bounds = outer_bounds;
        inner_bounds.Expand(new Vector3(-2*innerBoundsOffset.x, 0, -2*innerBoundsOffset.y));
    }

    void Update() {

        var displacement = target.position - transform.position;

        if (inner_bounds.Contains(displacement)) {
            bounds_reached = false;
        } else {
             if(! outer_bounds.Contains(displacement)) {
                bounds_reached = true;
            }
        }

        if (bounds_reached) {
            var dir = (displacement - inner_bounds.ClosestPoint(displacement)).normalized;
            var point = Vector3.Dot(dir, displacement)*dir;

            // choose 1 of the two:
            transform.position = Vector3.MoveTowards(transform.position, transform.position + point, camSpeed*Time.deltaTime);
            // transform.position = Vector3.SmoothDamp(transform.position, transform.position + point*2, ref velocity, 0.5f);

        }


    }

}