Indian indentured servants were common in Trinidad, Guyana, Mauritius, Fiji and South Africa and continue to make up a large percent of these countries population today. That said, it is important to keep in mind that these weren't slaves, but indentured servants. They generally worked on five year contracts. A common contract might be six months advance pay, and 5 rupees a month after that, plus food and board provided by the employer. While many of these indentured servants were hired by unscrupulous recruiters who lied about the conditions and renumeration they could expect, and it could be argued that much of the deperation of rural India was driven by British policies, recruitment to be an indentured servant was generally semivoluntary, and many indentured servants chose to stay in the countries they migrated even after the terms of their contracts ended. However, indentured servants differed from truly free workers in one important way. They generally had a contractual obligation not to change employers, even if they could get better terms elsewhere, nor could they return home until their terms of service ended. In many ways, the indentured servitude model never really ended, and the Kafala system in much of the oil rich gulf countries is very similar to the old indentured servitude model. While not exactly slavery, there were important elements of forced labor in indentured servitude and it was a deeply exploitative model of labor relations.

We tend to think of the Slavery Abolition Act of 1833 as having done just that: abolished slavery within the British Empire. It did not; at least not entirely, nor immediately.

Excepting Antigua, which chose to enact immediate and full emancipation in 1834, the Act enumerated gradual emancipation. The first step in this was conversion of the enslaved to apprenticeships - an act which still gave white society some control over the now "free" black population. This wasn't surprising at the time though as, at least in the British Caribbean, free blacks had long been required to enter into apprenticeships as a way of controlling their movement and labor. The only difference was that these apprenticeships were now enforceable for a set term (1840) rather than indefinite.

More directly related to your question are sections 64, 65, and 66 of the Act.

LXIV. And be it further enacted, That nothing in this Act contained doth or shall extend to any of the Territories in the Possession of the East India Company, or to the Island of Ceylon, or to the Island of Saint Helena.
LXV. And be it further enacted, That in the Colonies of the Cape of Good Hope and Mauritius the several Parts of this Act shall take effect and come into operation, or shall cease to operate and to be in force, as the Case may be, at Periods more remote than the respective Periods herein-before for such Purposes limited by the following Intervals of Time: videlicet, by Four Calendar Months in the Colony of the Cape of Good Hope, and by Six Calendar Months in the Colony of the Mauritius.
LXVI. And be it further enacted and declared, That within the Meaning and for the Purposes of this Act all Islands and Territories dependent upon any of the Colonies aforesaid, and constituting Parts of the same Colonial Government, shall respectively be taken to be Parts of such respective Colonies.

64 specifically limits the Abolition of Slavery within South Asia, 65 delays its implication in South Africa and Maritius, and 66 applies the status of the nearest administrative colony to surrounding islands and territories. This, in effect, allows the continuation of Slavery within South and Southeast Asia.

This then raises the question of "why?" And the simple answer is that it boils down to economic output.

The Caribbean sugar islands, which the Act was largely written for, had been unprofitable for quite some time by the 1830s. By the mid-18th century the boom of the sugar economy had steadily stagnated, and a series of upsets drove the islands to economic ruin. Among these were the loss of cheap provisions from the American colonies after 1775, the costs of garrisoning large military forces in the islands to stave off slave revolts, stagnation of infrastructure development as absentee-ownership steadily grew, and eventually the loss of price protections for sugar. By the early 19th century there was little economic impetus to see slavery continued in the British Atlantic world - and giving in to abolitionists there made sense. It allowed the abolitionists to a victory by doing away with slavery in the place that everyone most associated with its worst horrors, and it did away with the financial hurdles to manumission which had long required costly investment without reward from British investors. In fact, one of the key provisions of the Act was the establishment of a fund with which to compensate former-slaveholders for the loss of their enslaved property.

On the other side of the world, India had truly become the "Jewel in the Crown" of the British Empire in the early 19th century. British colonial authorities oversaw a campaign of deindustrializing India, and in turn making it the Empires largest provider of raw materials. These materials made it back to a rapidly industrializing Great Britain, where they were in turn manufactured into finished goods - which could then be sold to the massive, captive, colonial population of India. The end result was that India was responsible for approximately 30% of the UK's GDP by the 1830s.

At its peak, somewhere between 5 and 10 million Indians were enslaved under British rule - and held few of the legal protections which had been granted to Africans held in a similar state. It wasn't until 1843 that slavery in British Asia was done away with, which is when indenture began replacing it. I would take a much less benign view on indenture than u/gnikavar2 does, and say indenture was more akin to slavery than modern migratory agricultural work, but its not an area I've studied closely enough to give a strong rebuttal of.

Sorry I can't remember exactly which country was mentioned

Fiji.