By etymology the word Hindu refers to people living near Sindhu riverbed. So Hindus must have been in North India initially. How and when did Hinduism spread to southern India? Before Hinduism spread to s. India, what were the religions (if any) there? Did these people voluntarily convert to Hinduism?
This question assumes that Hinduism came from outside of India though Aryan "invasion". The AIT is disputed and increasingly losing credibility. When you make the assumption that Hinduism came from outside through invaders, you make the assumption that it spread like how Islam and Christianity spread whose ideology itself asks it's followers to proselytize others. Mapping Hinduism to this world view is incorrect.
Hinduism is not a monolith. It is a collection of "organic" cultures which originated and existed in India before colonization. There was always diversity and unity was never forced upon people.
Tamil people largely followed Saivam. Sangam literature is considered the oldest Tamil literature. First and second sangam literature is lost. But in the third sangam literature, it is mentioned that Shiva was the chief of the first Sangam and Murugan was the chief of the second Sangam. Nakkiran was the chief of the third Sangam and he was a great devotee of Shiva.
Saiva Siddantha tradition was developed in Tamil. It is considered non-vedic as the philosophy and methods of worship of Shiva are different from those in the Vedas.
Other than Saivam, there were lots of local deities and local heros like Madurai Veeran, Karupusaamy, Aiyanar, Sudalai Madan, etc., as well as animistic tribal traditions.
Vedic traditions supposedly spread to South India through sage Agathiyar who is considered the father of Tamil literature. The spread did not 'convert' non-vedic people nor did it try to eliminate them. Rather it created an organic unity. The above mentioned local deties were absorbed into the Vedic family without erasing them. That is, people continue to worship them the same way as they did before vedic influence. The only difference is a bond was created with the vedic deities. For example, Karupusaamy became the Vishnu's personal guard and a path to Vishnu.
The concept of "Aram" in Thirukural is similar to "Dharma" in the vedic traditions and sometimes considered to be the same.
Hence there was always a organic unity between various traditions in India, north and south and there was not 'conversion' or replacement of 'native traditions'.
UPDATE: See this paper Vedic Roots of Early Tamil Culture by Michel Danino which discusses this topic.
There is no question of conversions in Hinduism. Because it was not a religion at all. The word Hinduism itself was began to use in the recent past. It is actually 'Sanathana Dharma' and it is the culture followed in all over the India. More than a religion it is a way of life.(and what @Dharmaputhiran commented is exactly true)
'Hinduism conversion' - that is rare term indeed and 'voluntarily conversion' that is just your opinion without considering the facts.
Brahmanism(historic vedic religion/culture)is the one which eventually changed it's name to today's Hinduism. Over time, Brahmanism gradually became Hinduism.
The following description related to Ramayana reveals that Brahmans and temples were there in South India before ages.
In Treta yuga Lord Rama visited South India in search for Sita. He built a bridge 'Ramasethu' to SriLanka in which floating stones were used. Also it is believed that one of the south Indian state Kerala was created by Parasurama which was also in Treta yuga.
According to the legends, Lord Parasurama created the land between Gokarna and Kanyakumari. Lord Parasurama the sixth incarnation of Lord Vishnu was the son of Sage Jamadagni and Renuka. As a mark of repentance for Kshatriya Nigraha sin, Parasurama meditated at Gokarna and invoked Lord Varuna (the Lord of the Oceans). Parasurama asked him for a boon. To absolve himself of the sins he had committed, he wanted to donate some land to the Brahmins. There was no land available because he already donated the whole land he obtained by the 21 round Kshatriya Nigraha to Sage Kashyapa. Lord Varuna told Parasurama that he would give him as much land as he wished. He told him to fling his Parasu (axe) from where he stood at Gokarna. The land from Gokarna till the point where the axe landed would be given to him was the boon that Lord Varuna promised him. The throw of the `axe' from Gokarna to Kanyakumari created Kerala. Parasurama donated this land to the Brahmins and settled Brahmins there in 64 gramams or villages.
32 out of the 64 gramams are in the Tulu speaking region (in between Gokarnam and Perumpuzha) and the remaining 32 gramams are in the Malayalam speaking region(in between Perumpuzha and Kanyakumari) in Kerala. Those in Kerala listed in the Keralopatti, the narrative of Kerala history are
- Payyannoor 2. Perumchelloor (Talipparambu) 3. Alatthiyoor 4. Karanthol 5.Chokiram (Shukapuram) 6. Panniyoor 7. Karikkau 8. Isaanamangalam 9. Thrussivaperoor 10. Peruvanam. 11. Chamunda (Chemmanta) 12. Irungatikkootal (Iringalakkuda) 13. Avattiputhur (Avittathoor) 14. Paravoor 15. Airanikkulam 16. Muzhikkalam 17. Kuzhavur 18. Atavur 19. Chenganatu(Chengamanadu) 20. Ilibhyam 21. Uliyannoor 22. Kalutanatu. 23. Ettumanoor 24. Kumaranalloor 25. Kadamuri 26. Aranmula 27. Tiruvalla 28. Kidangoor 29.Chengannoor 30. Kaviyoor 31. Venmani and 32. Neermanna (Niranam)
So according to this, the land was created by Parasurama and later donated to Brahmins which clearly indicate that Brahmanism was in South India at least from Treta yuga according to this. Also it states that 108 Shivalayas(temples) were consecrated by Parasurama. You can check the list in the reference
Parasurama had consecrated 108 Shiva temples and 108 Durga temples for the well-being and prosperity of the people in Kerala . Among these 216 temples, the Lord Shiva of Gokarnam Mahabaleswara Temple in the north and Goddess Kumari of Kanyakumari temple in the south were considered as the protectors of Kerala. The first Shivalaya created by Parasurama was the Thrissivaperoor Vadakkunnatha Temple and the last one was the Thrikkariyoor Mahadeva Temple.
See the reference here
Hinduism did not spread to South India. IT is native to South and North India both. If you are talking about the relatively modern term Hinduism then you are mistaken even though the origin of this word is same as you explained in your question. The right name of Hinduism is Sanatan Dharma which has existed in South India from very ancient times.