For Pakistan, India and Saudi Arabia, visa processing is often the real constraint on how late you can book, not flight pricing. e-Visas for India and Pakistan are typically processed within 4 to 7 days, but delays happen, particularly during high-demand periods. Build at least 2 to 3 weeks of buffer before your travel date if you haven't applied yet, and treat "fastest possible visa turnaround" as a worst case, not a plan.
The order of operations matters as much as the timing. The expensive mistake is booking a non-refundable fare first and applying for the visa second, because if the application is delayed past your departure date, the fare is lost. If your dates depend on a visa you don't yet hold, either apply before you commit to the flight, or choose a fare with a change policy you've actually read rather than assumed.
Two document checks catch people out beyond the visa itself. If you travel on a NICOP or OCI card instead of a visa, check its expiry when you start planning rather than at the airport, since renewals routinely take longer than people budget for. And check your passport's remaining validity: countries on these routes commonly require six months beyond your travel dates, which turns an "expires next spring" passport into a problem for a winter trip.