I do domestic to international connections through T3 at Delhi Airport often.
While the answer by user DumbCoder is correct, it needs a bit more information to be complete, as mentioned by user Mophotla in the comments.
There are two scenarios to consider here: whether the first leg is a part of a longer international flight, or a domestic only flight.
In this scenario, you are flying from BOM to DEL in a flight that either originates or terminates at an international destination. An example of this kind of flight is AI101, that goes BOM-DEL-JFK. In this case, passengers travelling only on the BOM-DEL section of the flight (or those connecting forward for a domestic destination) will be given a boarding pass that has a big letter D on it. In your case, since you will be transferring to an international destination, your boarding pass will have a big letter I on it.
Since it's the same aircraft and boarding will take place from the same gate, passengers only travelling domestically will have to go through a quick immigration control, in which only the boarding pass is looked at. If this scenario takes place, in your case (with the letter I), you will have to go through exit immigration in Mumbai itself and only go through a transfer security check in Delhi.
In this scenario, you are flying from BOM to DEL in a flight that is solely a domestic flight. An example of this kind of flight is AI806 (as mentioned by user Mophotla). In this case, you will simply go through the domestic security in BOM, arrive in Delhi, and be redirected to the international gates. Here, you will first go through immigration and then transfer security. You will have to follow the signs for international transfers in Delhi in this case. There will also be Air India staff when you disembark to take you to your connection.
Tips and other info
- If you are in a premium cabin, please note that the transfer immigration and security counters at DEL do not have a priority lane.
- If arriving at Delhi around 10:00-12:00, you may have to face a long wait at transfer security, since 13:00-15:00 is a major international departure period for Air India and a lot of passengers are connecting at that point. Most Air India flights to Europe and some to SE Asia depart at that time from DEL. In my experience, domestic to international connections at DEL have been extremely quick every time except when I'm connecting for a flight going to Europe (that is, when my connecting flight is leaving in the early afternoon).
- If you have a big I letter on your boarding pass in BOM, you may face some issues with security not knowing which entrance you should use. In this case, follow the signs that direct you to the far left side of Mumbai's Terminal 2. This is the security check for premium passengers and those who have a big letter D on their boarding passes. Insist on going through this security and after the check, there will be an Air India representative sitting on the left side who will direct you to the right immigration counter (to make sure you go through exit immigration in BOM).
- If you are in an Air India premium cabin on your outbound flight from Delhi (or have status), you are also eligible to use other Star Alliance lounges in DEL. The Singapore Airlines and Lufthansa Group lounges are significantly better than the Air India lounge.