So, in Europe, fairly frequent mentions are made of a 'Campaign Season', delineated by when troops (especially feudal levies) could be raised, when the food surplus existed to feed a large group of men in one place, and when men were needed back home (usually) for the harvest. Often, this has a strong connection to the fall harvest and the onset of winter, with many early modern sieges needing to be abandoned because they dragged on too late in the season.
Did any equivalent exist in the Indian Subcontinent? My assumption would be that the rainy season would make roads difficult or impassable and the increased incidence of disease would make it hard to keep an army in fighting shape, but I've having a hard time finding definitive sources.
You are correct that Early Modern Indian armies preferred to campaign in the dry season.
The Mughal war season typically began in October, when the June-September rainy season gives way to the coolness and dryness of winter and the harvest season ends in the Arid Zones of India, the major centers of imperial military recruitment.
The South Asian subcontinent, today and especially in historical times, can be divided into two large ecological zones: the Monsoon and the Arid. The humid Monsoon Zones, most importantly Bengal, are marked by remarkable agricultural productivity in both of the two growing seasons, with transplanted rice the most important crop. In the more arid and hilly parts of India (like Rajputana), millet and wheat predominate. Both crops require less labor than rice but are much less productive. While the warm climate of South Asia can support two growing seasons (the kharif crops in summer and the rabi crops in winter; both the kharif crops were harvested and the rabi crops sown in September-October, the end of the rainy season), the peoples of the Arid Zone tended to focus on the kharif crops and leave many winter fields fallow.
This raises the question: how could people in the Arid Zone support themselves when their fields were lying fallow, especially with the reduced productivity of their homelands?
Some took on cattle herding, others weaving, and yet others entered the peddler trade. But the Indian peasantry was a very militarized population as a rule, especially in the Arid Zones. For many, the most obvious solution was to enter the military labor market as bandits and soldiers. It is no coincidence that Early Modern India's "three major military recruitment zones" that one historian identifies -- Rajputana, Maratha country in Maharashtra, and the uplands of the Nayak kingdoms in the subcontinent's southern tip -- are all part of the Arid Zones.
This meant that large infantry (and some cavalry) forces of militarized peasants could be raised in October, though most had to be disassembled the next year for the kharif crop.
Like most Indian empires, the Mughal state also relied significantly on these peasant forces. The conventional account of the Mughal army centers on an immense cavalry force of mostly Muslims and Rajput nobles, but actual local forces that maintained regional order, and probably a significant portion of Mughal armies on campaign, were from the ranks of the armed peasantry. An example: the chronicler Bhimsen, whose father was a clerk for the Mughal army in Aurangabad, Maharashtra, organized an army of Mughal footsoldiers recently raised from the distant Arid Zone areas of Narnala (232 kilometers to the northeast) and Bundelkhand (700 kilometers to the northeast) for the war against the Marathas.
The Mughal military machine needed food, transportation, and manpower. All three made it preferable for the war campaign to begin with the dry season. The lush kharif crops would just have been harvested and the countryside stocked with supplies for any lumbering army, while transportation was much easier in the dry season as you guessed. Finally, the end of the kharif harvest in the Arid Zone would free up tens of thousands of peasant-soldiers from agricultural labor and allow them to join the campaign for their share of the loot.
Autumn and winter were thus the Mughal war seasons.
On the other hand, the vast size of the Mughal empire meant that there was great variety in seasonal warfare. On the Persian frontier in the frigid mountains of Afghanistan, Mughal commander after commander found that snow and cold made beginning the campaign in October disastrous. On the other hand, in the humid riverbanks of Bengal, not beginning the campaign in the dry season was disastrous, not only because transportation was difficult but also because the more watery conditions allowed the enemy to deploy their powerful fleet of river-warboats to full effect. Yet in some places, the highly efficient Mughal supply network allowed sieges and campaigns to continue through the year. The last powerful Mughal emperor, Aurangzeb, famously had the audacity to besiege the city of Golkonda throughout the entire rainy season.
Sources