Yes, they are treated the same way as most general aviation aircraft, and yep, they do it all the time. International flight can be as short as from Buffalo, NY, across the lake to Toronto, Ontario, Canada, or from Arizona to Mexico, and a Cessna Citation X can fly from Detroit, Michigan to Toronto, Canada, or from Phoenix AZ to Mexico City. It does not even need to be a jet. Even you can fly internationally in a Cessna 172.
The smallest business jets are the Very Lights (VLJs) like the Hondajet and Phenom 100, and those aircraft have a max range of 1,500 miles (ish). I’m assuming by international you mean long trips across oceans, and the small jets won’t have life rafts or some cockpit equipment required to fly across the ocean, which is fine because their cabin about the size of a train compartment. Also there’s limited luggage room, potentially filled largely with survival equipment required for 100nm over water flights. From the US to Europe, we’d preferably go Bangor, Maine to Goose Bay to Keflavik, and from there it depends on final destination. In general, trips like this make sense only for ferrying or special purposes.
A private jet can fly around the world with careful planning, but due to the range of most small jets, multiple stops for refueling must be made and the route would have to be carefully planned not only for fuel and range but also to ensure that one is flying within the airspace of friendly countries. It totally depends on the capability of the airplane and the length of the route you are asking about, and whether they are permitted will depend again on the abilities of the airplane, the crew qualifications and on‑board equipment, and will they be permitted to land at the International airport.
In Canada, any general aviation aircraft with fewer than 15 persons aboard and no cargo or commercial goods is cleared through customs and immigration in the same manner. General aviation aircraft are cleared at their fixed base operator, and some airports have a dedicated customs ramp to clear private aircraft. Between 2 and 48 hours prior to arrival, the pilot in command will call the Telephone Reporting Centre with aircraft make, model and registration, last airport of departure, airport of arrival, estimated arrival time, passenger and crew manifest, and declarations. If the aircraft is released, everyone is free to disembark, and if the aircraft is referred for inspection, two or more officers will be dispatched to attend the plane.