I can't really talk about the Mughals' reaction to the caste system, but rather about its modification in Colonial times, which hopefully will shed some light on earlier times as well.
In a recent article, Dharampal-Frick argues that European, specifically British understandings of 'race' strongly influenced and later restructured Indian caste, traditionally known as varna:

Paradigmatically, these varnas […] constituting merely conceptual social categories, were understood […] to represent actual social groups. […] This integrative ritual scenario in which Hinduism abounds, signifying from an emic Indian perspective the single racial origin of all social groups, represented primarily the embodiment of a holistic, organic vision of human community. Yet this metaphorical conceptualisation was taken literally by the early Orientalists as designating the ranked functional and religiously sanctioned hierarchy of the Hindu body-politic.

Thus, from the late 18th c. onwards, British administrators and scholars interpreted scriptural varna dictates through race-based frameworks, facilitated by the new science of anthropometry (physical measurements being connected to supposedly 'superior races') and by 'Orientalist' appropriation of selected Indian scriptural traditions (as with law texts) – Accordingly, service personnel were incorporated into the civil administration (so-called 'upper castes') and the military machinery ('martial classes'), and rural communities ('criminal tribes') were marginalized and at times even exterminated. These processes gained momentum especially after the massive 1857 rebellion against British rule and with the subsequent restructuring of administration, government and military.
Some western-educated Indians deployed the superimposed colonial categories to further their own goals towards equality, based on a supposed Aryan kinship. Others who had become disempowered used the racial vocabulary to formulate ideas of subalternity and victimhood vis-à-vis the brahmanised/north Indian and western-educated groups, leading to societal schisms and political conflicts, as well as influencing later misunderstanding regarding the definition of caste.

So, what has this to do with the Mughals? The view of pre-colonial varnas as representing “merely conceptual social categories” would seem to make them more of a part of spiritual rather than political life. As the Mughals, similar to other Muslim rulers, allowed for religious liberties (usually connected to special taxes payed by non-Muslims), these customs would not have directly interfered with Mughal rule and political institution-building. Furthermore, Mughal rulers like Akbar are known for the atmosphere of cultural freedom at their courts, with Akbar ordering a translation of the Koran in Persian, and for incorporating local elites from various religious groups in their administration. Hopefully someone with more knowledge of the Mughals can add to this.

To sum up: In the British Raj, the traditionally religious concept of 'caste' was strongly influenced by orientalist scholarship and restructuring of the administration, leading to its political connotations until today. Mughals often worked together with local elites and patronized culture, so that the non-political concept of 'caste' might have simply been accepted as part of indigenous spiritual customs.

Sources: Gita Dharampal-Frick: Interrogating the Historical Discourse on Caste and Race in India, in: History and Society – New Series, New Delhi 2013, as well as a few of her lectures.