The spread of the Indo-European languages from Ireland to India, and their connections to the third wave Steppe riders is best explained in this book by David W. Anthony, "The Horse, the Wheel, and Language: How Bronze-Age Riders from the Eurasian Steppes Shaped the Modern World published by Princeton University Press, First published July 26, 2010
Wikipedia has a thorough analysis of the book.
Below are some images from the book which may also be purchased from Amazon and Barnes and Noble.
The thesis of Anthony's book is seen in the title. The domestication of the horse as a draft animal led to wheels and carts which made existence on the Eurasian Steppe possible. The language that was developed is now called Proto Indo-European (PIE) and Anthony (and these pictures) focus on words related to horses and wagon. He chooses five key words for parts of a wagon (two for what we call wheel and rotor) and saw that not all languages had all five. He believes (and it is now generally accepted) that this happened in what we call the Pontic-Caspian Steppe which denotes the plains in southern Ukraine and Russia above the Black and Caspian Seas. (Pontus was the Greek word for the Black Sea.) We call these people the Yamnaya and we share much of their DNA.
The Tocharian people are an early branch of PIE which went east and disappeared from history, but traces of their language remain. They may be related to the fair skinned, plaid wearing Tarim Mummies who were found nearby. A major sub-group of Indo-European is called Indo-Aryan and gave rise to the Iranian (Aryan) and Indic languages. Iranian is spoken in a large swath from Kurdistan through Iran, and Afghanistan. Indic gave rise to Sanskrit and its daughters Hindi and Urdu spoken in India and Pakistan respectively. The Anatolian (modern Turkey) branch has also disappeared but has an important history. It was spoken by the Hittite Empire and the people of Troy who fought another Indo-European people in the Trojan War.
Researchers have been working for years to understand PIE and have built a dictionary of over 3,000 words which help us understand their religion, society, and activities. This helps us develop a timeline for the separation of the modern languages from PIE throughout Eurasia. In the third image below, we see the western branch composed of Germanic, Italo-Celtic, and Balto-Slavic where Baltic refers to Lithuania and Latvia. So, the Italic languages were the last to separate from Irish which means that apart from the other Celtic languages, Italian is the closest to Irish.
Unlike other European languages which have an Old, Middle, and Modern version, Irish also has an Archaic which is similar to the Italic languages including Modern Italian. In MacNicol's book, he states:
If the Cunagusos whose Ogam stone was erected in East Cork had lived in the seventh century rather than (perhaps) and the fourth, his name would have been Congus;
were he a poet, he would have been not a velitas, but a file; and his sister would have been not his swesur, but his siur: three examples illustrating the fact that
between the fourth and seventh century, Irish underwent a series of drastic transformations.
Of these the chief were introduction of lentation, the dropping of final syllables, elimination of certain internal syllables, the reduction of unstressed long vowels, and the reduction of certain consonant groups with consequent compensatory lengthening short vowels: formidable list, which in effect means that the primitive Irish of the fourth century, but a time that evolved into the old Irish of the seventh century, was recognizable only with difficulty.
Of particular interest is the name change of the capital of the Uladh (Ulster) and their principal tribe, the Dál Fiatach. We know it as Emain Macha, but Ptolemy's Map uses the old Celtic form Isamnion (Byrne p50). Ptolemy also calls the name of point of land into the sea in County Down, Isamnion Point. This is very near to Downpatrick the base of the Dál Fiatach.
We know of Archaic Irish only from ogam stones which were monuments to important figures and inscribed in Archaic Irish. Ptolemy lived in Alexandria Egypt, never saw Ireland, and got all his information from others. What is notable about the map is the lack of any mention of the great O'Neill kingdoms or of the Laigin (Leinstermen). Thomas Francis OʼRahilly, who was a great historian of the last century, beleives Ptolemy used the work of Pytheas of Massalia who visited Ireland around 325 BC. Ptolemy's map shows an Ireland dominated by the Eireann, not the Gael. It includes the name of Belgic/British tribes, not seen later. The term Gael was coined by the O'Neill to refer to themselves (Feni was the people, Gael was the ancestor). later it was applied to all Ireland to enhance O'Neill control. The Dál Fiatach and the Dál Riada considered themselves Eireann and they are both on the map.