JohnBurke

Craig,

Scribophile is a great place for discussions related to character, plot, or specifics, and there are always a few participants.

Flight plans are filed in order to interact with air traffic control, which is required above certain altitudes (which vary by country). Turbojet aircraft need to fly at higher altitudes for efficiency (fuel consumption/speed), and travel everywhere interacting with air traffic control, and on instrument flight plans (IFR, or instrument flight rules).

Associated with filing a flight plan on international trips is obtaining overflight permission over each country. Most countries will also have a fee for passing through their airspace and using their air traffic control services, which will be billed to the aircraft owner or operator.

Oceanic trips over heavily travelled routing, indeed most routing, involves additional services in which the flight is monitored by one of several means, rather than actually watched on radar (too far away for radar in most cases). Many areas used for oceanic crossing are dense in air traffic, requiring a specific clearance to fly a track or route. In the north Atlantic, for example, one must fly one of a number of tracks, passing over each waypoint within three minutes, maintaining a specific altitude. Arbitrarily trying to sneak through would be dangerous, and would likely be noticed at multiple points as traffic collision warnings sound on various aircraft, as well as the exit and entry from radar control.

Unknown aircraft in most countries are defense concerns. In some countries, they will be intercepted.

If the antagonist in your story is smuggling children, they're better smuggled as cargo than as patients; a patient still needs a customs declaration upon entry and identification.

As others noted, your aircraft will require two (or more) crewmembers.
More secrets to keep, though it's possible that the crew doesn't know the purpose of the trip. Charter flights often don't. I've flown a lot of people places with no idea of their purpose; that's often why people charter. Privacy.