What you describe sounds like Ease-in-sine, like so:
LeanTween.move(this.gameObject, new Vector2(0f, 5f), 2f).setEaseInSine();
This moves the gameobject the way you describe it: Slowly starting, accelerating and abruptly stopping.
When typing ".setEase", your IDE should suggest you ways to complete the code (See Screenshot). There you will find many different presets of easing-curves. Like only easing in, only easing out, or both combined.
If you are not sure what easing to use, you can look at a collection of some of the most used easing-curves here: https://easings.net/en
Like @Lotan already suggested, creating your own animation curves would give you the ability to fully customize the easing behavior.
