Is there any special clearance, when military aircraft will overfly other countries' airspace? Are they continuously monitored?
In general, a diplomatic clearance is required for overflight of military aircraft over other countries. The aircraft are considered state aircraft and prior clearance has to be obtained through diplomatic channels. Note that the clearance is usually issued by the defence ministry of the concerned country.
The mode of clearance and monitoring varies from country to country. In case of US:
Foreign governments seeking diplomatic clearance for state aircraft to transit U.S. territorial airspace or land in the United States must obtain a diplomatic clearance number (DCN) issued in advance by the Office of International Security Operations, Bureau of Political Military Affairs (PM/ISO), Department of State. The DCN authorizes the aircraft to transit or land in the United States and/or its territories, in accordance with the approved itinerary.
The overflight permissions are also dependent on agreement between various countries, like India-Pakistan. In some cases, like some EU countries, military transport aircraft can operate without diplomatic clearance.
Some other countries, like Canada give blanket clearance for military aircraft from some countries:
3.2.3 Blanket Diplomatic Clearances to Foreign Military Aircraft
By order-in-council, United States, United Kingdom, and New Zealand military aircraft are exempted from the requirement to have a formal diplomatic clearance (see 3.2.1 above) prior to overflying and/or landing in Canada.
Military aircraft from North Atlantic Treaty Organization countries other than the United States and the United Kingdom are granted clearances on an annual basis. These clearances do not apply to aircraft carrying armed soldiers, armaments and/or other dangerous goods, nor do they apply to aircraft carrying government personnel.
The procedure is usually the same in most of the countries- the overflight request is given through diplomatic channels to the countries' defence ministry, which then evaluates it and approves (or rejects it). The procedure for Canada is given here:
The state of origin will provide full flight details, as outlined in Annex A, in a diplomatic note from its embassy to Foreign Affairs and International Trade Canada (DFAIT). DFAIT will consult with the Department of National Defence (DND) Operations Centre and, if the foreign military aircraft is landing at a civilian airport, with the Foreign Inspection Division to determine whether any pre-conditions should be imposed. After this consultation, DFAIT will reply to the requesting embassy by a return diplomatic note. On advice from DFAIT, the DND Operations Centre officer will then issue the appropriate flight clearance and distribute copies to the interested parties.
In case of India:
For clearance of flights by foreign military aircraft, the concerned diplomatic mission in India or abroad are required to apply to the Ministry of External Affairs with copy to Air Headquaters/Naval Headquaters forclearance of such flights. After clearance given by Ministry of External Affairs, Air Hqrs/Naval Headquaters will issue necessary AOR number for operation of such aircraft, whether it is overflying Indian air space or landing at any Indian airport (civil or defence).
The monitoring of overflying aircraft depends on the country. Civil ATC usually monitors the aircraft carrying transponders, while the military radars (where available), monitor all air traffic and scramble fighters when the aircraft is not identified/responding properly. Overflight without permission may at the very least result in the aircraft being forced to land by authorities.
In Switzerland, every aircraft is monitored by the air traffic control (ATC). However, if the plane flies without a transponder, it's an unknown aircraft for the ATC-Radar and the military will take the necessary steps to identify the aircraft. Head over here for more details.
At the WEF (World Economic Forum), politicians from all over the world visit Davos. Most of them fly with jets from their air force. Look at this question about the Boeing E-4B Advanced Airborne Command Post 'Nightwatch', carrying Ashton Carter to Switzerland.
Foreign state flights over or into Switzerland require special permission, known as diplomatic clearance. This permission is granted by FOCA in consultation with other federal offices.
State flights include flights engaged in missions undertaken by government, military, customs and police authorities. They particularly include flights by order of the state, flights used for transporting heads of state and high-ranking government officials on official duties, flights used in police and customs operations, and all military flights.
You can find more information about this in the link provided. Keep in mind, this applies only to Switzerland.
These flights often get intercepted and identified by the swiss air force. Here is a video showing two F5E Tiger II intercepting the Canadian prime minister as he leaves Swiss air space after the WEF 2016