Yes. There were certainly economic migrations in Asia during the late medieval period. Anywhere you find major cities there were migrant communities of merchants, artisans and intellectuals. Long distance labor migration was less common than today, mainly because most people in this period worked the land. It is only in the industrial period where people have migrated from peasant societies to cities that we typically see these long distance permanant moves. There were frequent movements of pastoralists and farm workers, but these were typically seasonal and circular (they moved then returned to the place of origin).

By immigration for "pleasure" I am not sure what you mean. Not many immigrants today make permenant migrations purely for pleasure. If you mean something like tourism - this was not nearly as easy or frequent as today. But many people did travel for Pilgrimages which would be the closest equivilent.

There were also many foreigners in the military. This was certainly true in India (and also Europe), although these were a mix of what historians have called "adventurers" and slaves. It was common for Islamic rulers from India to North Africa to employ slaves in the military and many rose to positions of significant power this way. In fact there were slave dynasties in both India (e.g. Biajapur Sultanate) and Egypt (Mamluks).

In India specifically there was also long distance internal migration. At the time India was split into numerous rival states, so these migrations were between separate territories. Brahmins, Islamic scholars and Bhuddist monks were all fairly mobile.

Sheikh (2010) talks about immigration to Gujarat in this period:

“From the twelfth to the end of the fifteenth century and even later, Gujarat was still in the process of being ‘settled’. The coastal towns were particularly prosperous and boasted sizeable and entrenched colonies of local and itinerant merchants. In the hinterland, however, migration and mobility were the norm for all those engaged in trade, politics, and pilgrimage.”

“Artisans and traders flocked to the manufacturing towns of eastern Gujarat from the eleventh century onwards. Traders and brahmins from Śīmāla in southern Rajasthan began to enter eastern Gujarat at this time. Many of them achieved prominence as priests, bankers, and merchants. Brahmins came from even further afield in search of patrons. The Kaṇbī peasants have a tradition of migration from Punjab. Within Gujarat they settled along the major trade routes and moved into the new towns such as Adalaj, Champaner, and Ahmadabad. ‘Foreign’ traders settled in the coastal ports in this period, particularly at Cambay.”

Sheikh, S. (2010). Forging a region: Sultans, traders, and pilgrims in Gujarat, 1200–1500. Oxford University Press.