History of the Ḍom People (ancestors of the Roma) within the South Asia (and partly beyond)

The original Ḍom People started migrating from southern India (apparently somewhere in between nowadays Tamil Nadu and Kerala, with stops taking longer time than 300 years in nowadays Andhra Pradesh, Telangana and Karnataka) to northern India in 2500 BC.

Some etymology to the name Ḍom.
It´s presumed root, ḍom, which is connected with drumming, is linked to word damara and damaru, Sanskrit terms for "drum" and the Sanskrit verbal root डम् ḍam- 'to sound (as a drum)', perhaps a loan from Dravidian, e.g. Kannada ḍamāra 'a pair of kettle-drums', and Telugu ṭamaṭama 'a drum, tomtom'.

So, Ḍom literally means the "drummers" = musicians.

Why Ḍom are ancestor of the Roma?
From Ḍom arose our ethnic name Rom - the Roma, in some dialects still with retroflexed R.
In elsewhere pronounced as Dom/Domari with normal "D" in the name (within the Middle East & in Pakistan) and Lom (in Caucasus).

Closest people to the original Ḍom are nowadays Adivasi tribes of the southern India, like the Irula, Paniya and Kurumba, apparently.

The nowadays Ḍom are nowadays seen as Indo-Aryan people. While the Ḍom (de facto) caste is often referred as one people, there are different Varnas and Jātis among them. 

Some extra theories by me:

Why do Roma are genetically (predominantly) and linguistically (fully) Aryan and not Adivasi or Dravidian?

After migration to northern parts of the Indian Subcontinent the Ḍom did two things:

There was inmix with the local Indo-Aryans (before it was forbidden by Hindu texts like Manu Smriti), as well with intermixing the northern-Indian tribes who were not mainstream Vedic-like people (i.e. nomadic tribes of the North-West India).

Sanskritisation (i.e. the shift from Dravidian, or possibly Adivasi languages and culture to the Sanskrit language and Brahminical/Vedic culture and social norms) of the Ḍom ancestors who became later the ancestors of the Roma/Domari/Lom.

Many Ḍom nowadays are atypical Jātis of the Varna system, but still are traditionally clean (or most of them), for example the Ḍom Bania from Bihar, who do the jobs as the Bania do, i.e. trade.

While there were some Ḍom who were culturally not Sanskritised to such extend and became totally marginalised, and as well some of the Ḍom started to do the ritually unclean jobs (like the Ḍom Raja from Varanasi who cremate the deceased on the Ghats and often leads the researchers to the conclusion that all the Ḍom are Dalti = Untouchable), as they got outcasted jobs, i.e. they became Dalit.

The heavy inmix with the local Indo-Aryans and non-mainstream Vedic-like people had huge impact on the genetics of nowadays Ḍom and via that as well the Roma, Domari and Lom people, beside cultural impact. (They became a part of the Varna system, i.e. culturally Aryan. Aryan is a cultural term, not race.)

It as well seems that the Ḍom had some autonomy with own rulers earlier, as there were reported the ruins of the Ḍomdigarh (Ḍom's fortresses') in eastern portion of nowadays state of Uttar Pradesh and according to some local myths the Ḍom People had a Ḍomraon city, where the Ḍom kings reigned from. And as well in the Buddhist Tantric literature is mentioned some Ḍom king named Heruka. (I must to mention that the Ḍom Raja = Ḍom Kings)

Notes:

Also, I have to mention, that not all the Roma, Domari and Lom are totally the same. Among the Roma are tribes, castes and clans, who have some genetical differences with a few or a lot cultural differences. And the Domari are also divided into tribal unions, and into castes apparently. About the Lom I can't say much.

Another thing, I must to mention is that migration waves outside of the Indian Subcontinent of the ancestors of the Roma, Domari and Lom occured in different periods of time, caused by many reasons, be it social reasons, invasions etc.

The biggest migration wave of the ancestors of the Roma apparently occured after the Hun (Hephthalite or Alchon Hun) invasion of the Gupta Empire, in 6th Century, with small migrations in around 300 AD and in times of Islamic Conquests around 900-1100 AD.

Some parts of the Roma in Sassanide Empire became militarised and behaved similarly like the Kśatriya Varna in the Indian Subcontinent. This military tradition is seen among tribes like Russka Roma, Servy tribe in Ukraine and in smaller portion among the Kalderash tribe of Vlax Roma. But I must to mention that there are some Adivasis and Dalits with the warrior traditions, so this doesn't points to that the ancestors of the Roma were the Kśatriya.

The Roma also tried to create own country in Iraq in times of the Abbaside Caliphate, when they led the Zotti (Zott was and is the exonym to call everyone out of India, as the Arabs knew Jats of Sindh first) uprising against Islamisation of the Non-Arab (and Non-Semite) People. (Before the Abbasides there ruled the Umayyad Caliphate and they were not doing forced Islamisation, especially among Non-Semite People, but the Abbasides believed that everyone must become a Muslim).

After crushing the Zottistan around 800 AD, some of the Roma fled to Armenia, Georgia and bigger part to the Byzantine Empire. It also seems that a lot Roma were also kept in some provisional prison in nowadays Syria and the Byzantines freed these military men, as many of the Roma at some point of history were in the Byzantine Army.

The vast-majority of the Roma seem to have little bit different ancestral sources than the Domari and Lom. In Domari is a small exception with the Banu Sassán Domari groups who seem to be originally a part of first Zott migration, which covered the Roma too. In fact the Banu Sassán are those who decided to move towards the east and not west, and fought on side of uprising in Persia against the Muslims, but later became Muslim themselves.

PS 1:
Not all "gypsies" in Europe are the Roma.

There are indigenous European "gypsies" who have not Indic blood at all, for example the Pavees (Irish Travellers) and Yéniche.

As well, there are some "gypsies" who are from Indic origins, but are not Roma, for example the Rudari/Beás, who also stand outside of the Romani cultural norms and castes and are seen as outcastes. Note -> some of the Rudari/Beás undergone historical Romisation and became partly a part of the Roma culture and adopted some norms and language, which gave them status to be seen as Rom among some Roma.
Note 2 -> some of the Rudari/Beás are not actually Indic, as they were taken by the Mongols in Central Asia.

PS 2:
Some of the "gypsies" in Eastern Europe are not Indic, nor indigenous Europeans. For example in Romania are some "gypsies" of the Middle Eastern origins, these were slaves brought from the Arabs/Turks from Syria mostly by the Orthodox Church of Romania. Some of these were probably Romised (same as some Rudari) and influenced some genes among the Roma. (That´s why some Roma get on Autosomal DNA Northern Levant)

PS 3:
Some of the Domari, under Banu Sassán tribal union came to the Europe in probably 14th Century and underwent Romisation and became a part of the Roma, they're known today as Gurbet tribes.

The Banu Sassán may intermixed with some local populations from Persia, and as the Gurbet tribes were (re-)Romised in Europe they maybe brought some additional Iranic genes to the populations of the Roma.

PS 4: The Roma also practiced adoption of the Non-Roma children. These were often abandoned and many people knew where are the typical routes used frequently by the Roma, so they'd put the children on these routes. The Roma adopted them and raised them as the Roma of the particular Caste (or better use Indic term Jāti) of the group which adopted these Non-Roma children. This is why the Roma have a large portion of genes of Non-Indic origins, including the Haplogroups.