This is an interesting question. Thanks for asking.
You will have to remember that after the death of Mongke Khan in 1259 CE, Mongolia declined in importance. This was because there were other centres of power that arose instead. Qaidu, the Chagatai Khanate, and the Golden Horde each, in turn, created their own centres of power or they continually ignored the role of the Khagan. As the result Mongolia essentially reverted back to the pre-Chinggisid era and was ignored, for the most part, by other Chinggisids.
By the mid of the 15th-16th centuries we see the rise of the Oriat, a Mongol Khanate, within the region of Mongolia. These Mongolians were not Chinggisids, though they did contain some Chinggisid tribes who joined them primarily from the Chagatai Khanate.
The only Chinggisid Khanate which the Oriats had contacts with is the Late Chagatai Khanate which was reduced to the regions surrounding East Turkestan. The only sources I know which deal with the Oriats is found in two Persians texts, written by Moghul themselves, Mirza Haydar Dughlat's Tarikh-i-Rashidi and Shah Mahmud bin Mirza Fazil Churas's Tarikh-i-Churas. The Tarikh-i-Rashidi has been translated into English and the Tarikh-i-Churas has been translated into Russian in the 1970s. Both of these texts detail, to some degree, the constant unsuccessful wars fought by the Chagatai against the Oriats primarily during the reign of two Khans.
During this time Islamization in East Turkestan was in full swing and it is clear at least in the Tarikh-i-Rashidi that the wars being fought against the Oriats were taking a religious overtone. The first Khan who fought with the Oriats was Ways Khan (r. 1418-1429) who is reportedly said to have fought 21 battles against the Oriats and only won once. The wars that were fought in his reign against the Oriat was because the Khan is claimed to have been extremely Muslim-minded and had forbidden the Moghuls to raid Muslims thus the Oriats were the natural target. During two wars he was captured twice by the Oriats and met Esan Taishi, the Khan of the Oriats, who calimed that
"If he really is a descendant of Chinggis Khan, he will certainly not bow to me but will look upon me as a liege man." When the Khan was brought in Esan Taishi was mounted. He dismounted and went forwards respectfully to greet the khan, who averted his face and did not extend his hand. This resulted in Esan Taishi having great respect for the khan, and he did the khan much obeisance and gave him leave to depart. (Tarikh-i-Rashidi, p. 35)
which indicates, at least that there was a clear differentiation that was understood between Chinggisid Mongols and non-Chinggisid Mongols. Unfortunately, I do not know of a corresponding Oriat source through which we could further examine this account. Another battle resulted in the capture of Ways Khan and the marriage of his sister to Esan Taishi. There is also account that Esan Taishi had converted to Islam and their sons were named Ibrahim and Ilyas but again since I'm not familiar with existing Oriat sources I'm not able to confirm whether Mirza Haydar was correct. However, he states that they eventually lost a power struggle against Amasanji Taishi and fled to China.
The second account of the Oriats occurs during the reign of Yunus Khan (1416-1487 CE) in which only one battle is mentioned against Amasanji Taishi in which Yunus Khan, with his Moghuls, left the Aksu region and migrated into Moghulistan (East Turkestan) after which there is no further indications that there was much contact with them further.
What I do know is that Muslim traders continued to ply their trade and that Esen Taishi had used them as well to trade with Ming China. I am not sure if he was overly familiar with Muslim culture but considering the presence to traders, Chinggisid tribes who converted and joined them from the Late Chagatai Khanate, and his relations with Ways Khan, including his supposed conversion, that he was familiar with Muslim culture. It is also clear, with available sources that Chinggisid Mongols were considered different and seen as being "better" than non-Chinggisid Mongol groups such as the Oriat.
I've listed some sources here you may find interesting.