The first thing to note is that the originators of the Aryan migration theory (like Max Mueller) had only linguistics to go on i.e. similarities between Sanskrit and European languages. And there is no doubt that their hypothesis was ideologically motivated. But that doesn’t say anything about whether it is correct or not. As it happens, the latest DNA evidence overwhelmingly supports the idea of migration from Central Asia into the Indian subcontinent within the last 5000 years.

I would like to base the rest of my answer on a study called “The Genomic Formation of South and Central Asia” published in April 2018. It is based on sequencing of ancient and present-day DNA and is published by a team of 90+ authors including archaeologists, many of whom are Indian. Please note, the genome study does not refer to any “invasion” but rather to migration over centuries.

Key findings from the study, in the terminology of the authors (also note, they don’t use terms like “Aryan” and “Dravidian”) -

1. The people who inhabit the Indian subcontinent are a mixture of 2 ancient populations ANI (Ancestral North Indians) and ASI (Ancestral South Indians). This was already described in previous genetic studies.
2. The ANI are a mixture of three, more ancient, populations - Iranian agriculturists, South-Asian hunter-gatherers and Steppe pastoralists. The ASI are a mixture of the first two but not the third one. Note that South-Asian “hunter-gatherers” discovered agriculture independently of the Iranian agriculturists (based on the latest study).
3. The South Asian hunter-gatherers were the first to arrive in the subcontinent from Africa long before any other population. As such, we can treat them as the “indigenous” population of India.
4. The Iranian agriculturists arrived between 4500 and 3000 BC. The Steppe pastoralists arrived later than 2000 BC.
5. What we call the Indus Valley (or Harappan) Civilization was comprised only of Iranian agriculturists and South-Asian hunter-gatherers. Steppe pastoralists arrived when IVC was already in decline.
6. Certain Indian population groups have dominant Steppe pastoralist ancestry
7. What we call “Aryan” population corresponds to Steppe pastoralists. They were the originators of the Indo-European languages so termed because of the similarities between Sanskrit and European languages.

Aryans are best defined as a linguistic group and not a “race”. Race is not even a well-defined concept in genetics. The same Steppes people also migrated to various parts of Europe, which is what explains the similarities between South Asian and European languages.

So the short answer is this - the Aryans came to India from Central Asia corresponding to modern-day Turkmenistan, Uzbekistan and Kazakhstan, between 2500 and 1500 BCE.

Post-script: To cover the arguments other than genetic studies I can do no better than to quote the foremost scholar on this subject Dr. Iravatham Mahadevan -

There is substantial evidence that the Indus Civilization was pre-Aryan.

1. The Indus Civilization was mainly urban, while the early Vedic society was rural and pastoral. There were no cities in the Vedic period.
2. The Indus seals depict many animals but no the horse. The horse and the chariot with spoked wheel were the defining features of the Aryan-speaking societies.
3. The tiger is often featured on Indus seals and sealings, but the animal is not mentioned in the Rigveda

If you would like to know a little more about how genetics has established that Indian subcontinent is descended from two different ancestral populations, please see here. There is also an excellent Quora answer by Ram Senthur that surveys all the available genetic studies as of Sept 2017. I can also recommend the book Early Indians by Tony Joseph.