Within Dharma, there is no process of ‘converting’ someone to Hinduism.
Everyone is considered to be born as someone who is doing “Dharma-acharana.” You might get lost along the path, but that is a different story.
Depending on a combination of your
1. janma jātī
2. janma matam
3. janma kula devī/devatā
4. janma kula ācārāṇi
5. varṇam
6. sādhanā sāmpradāyam
7. guru paraṃparā
8. sādhanā matam
9. sādhanā kūlam
10. sādhanā upa kulam
11. sādhanā paddhati
12. sādhanā mārga/paṃthī
in that order, you can belong to a very nuanced, complex combinatory sytem of rituals and worship.
Someone either born into this or ones that are lucky enough to find a Guru, never ‘convert’ or are ‘formally welcomed into’ Dharma, as we see with rituals such as Baptism in Christianity.
But, then there are some saṃskāram rituals that you can perform for someone who has gotten lost along the way. These have been documented in two sources.
The first one is the Devala Smriti written by Sage Devala. He is the sage of the weavers. He was the first weaver in the Dharma timeline and is considered somewhat of a patron saint to the tantuvāhī/devāṃganā community in India.
In the smriti that he wrote, he has given a list of vows that can be taken in front of agni, while your community leader or guru is present at the spot, to ‘rid you of your past-sanskaras’ (often wrongly translated as purification) that have come about due to your ‘past-life.’ It is a ‘re-birth.’
Although this sounds quite similar to what Dwijas go through, where they take up yama-niyamas and go through a rebirth, the process and even the whys and hows are completely different.
The second source is not an ancient one, but a modern one. It is an agni kriya ritual designed by Swami Dayananda Saraswati, who founded Arya Samaj.
But now, 2 questions come to our mind, those who are born into Dharma, have a janma jātī, janma matam, janma kula devī/devatā, janma kula ācārāṇi. But for those who have newly taken up the vows, how do we decide their jātī?
As varṇam is something that a person ‘chooses’ there are no problems there. But what about jātī? The one due to whom you came back to Dharma and took/re-took these vows, their jātī becomes your jātī.
All these issues arise, only if you are concerned about jātī which the Arya Samajis are not concerned with. They only accept varṇam and not jātī.