It was because from the very beginning the Chagatai Khanate was split to 2 countries.
They were Western Chagatai, and the other was Eastern Chagatai.
I don't know why the English Wiki quotes this incident so minimally, only saying,
The khanate lasted in one form or another from 1220s until the late 17th century, although the western half of the khanate was lost to Timur's empire by 1370. The eastern half remained under Chagatai khans, who were, at times, allied or at war with Timur's successors, the Timurid dynasty.
It was not according to the "fall", but by the replacement by Timur.
So where did Eastern Chagatai Khanate go?
Here is the summary.
Moghulistan (Mughalistan, Moghul Khanate) (from Persian: مغولستان, Moqulestân/Moġūlistān), also called the Eastern Chagatai Khanate (Chinese: 东察合台汗国; pinyin: Dōng Cháhétái Hànguó), was a Mongol breakaway khanate of the Chagatai Khanate and a historical geographic area north of the Tengri Tagh mountain range,1 on the border of Central Asia and East Asia.
And the same Wiki says,
In actuality, local control rested with local Mongol Dughlats or Sufi Naqshbandi in their respective oases. Although the rulers enjoyed great wealth from the China trade, it was beset by constant civil war and invasions by the Timurid Empire, which emerged from the western part of the erstwhile Chagatai Khanate. Independence-minded khans created their own domains in cities like Kashgar and Turfan. Eventually it was overcome by the Kyrgyz, Kazakhs, and Oirats.
So after the death of Chagatai Khan, the "dynasty" was split and had to fight with another normad people, so I think there is no afford for them to invade India.