What is the procedure behind granting political asylum to people at embassies? (mainly how can people enter an embassy, why don't police arrest them, how do people stay at embassies) Please cite any law or rule regarding it and also the explain the extradition procedure in such situations. Also how long is a person granted political asylum?
Entering an embassy is normally easy, you just go through their security measures at the door and enter. (It may be difficult for a fugitive to travel to an asylum, but that is beyond the scope of the question)
Claiming asylum in an embassy is rare, It is sometimes seen in China that dissidents will ask for protection at the US Embassy in Beijing. On entering the embassy you ask for protection. The ambassador must make a judgement based on the prevailing human rights laws, "The International Covenant on Civil and Political Rights", If they judge that you have a reasonable risk of being killed or harmed if you are handed over to the host authorities they have a duty to keep you at the Embassy. Note that "asylum" is not granted at the embassy, only "protection".
Under established diplomatic convention, the host country's authorities cannot enter the grounds of the embassy.
The Ambassasdor would then consult with her superiors to decide what should be done next. The embassy will usually want to get rid of the person as soon as possible, as their presence is politically awkward. It is likely then that they will consult with the host country to try to guarantee fair treatment. Or arrange a deal with the host country to allow for the person to leave under the protection of the embassy.
There is no real extradition procedure, as the person hasn't left their country. The embassy only has to satisfy itself that the person won't be killed or tortured, and then they can ask the person to leave. If there is no deal in place, then the person is likely to be arrested as they leave.
If the embassy isn't satisfied that the person will be safe, the person may remain in the embassy for a long period of time. Jozsef Mindszenty remained at an embassy for 15 years.
Consider the cases of Fang Lizhi, who entered the US embassy in 1989 during the Tiananmen Square massacre. He was allowed to remain for a year, The Chinese then did a deal in which Fang was allowed to leave China to join the University of Cambridge, and later he live in the USA.
On the other hand Manuel Noriega tried to take refuge in the Vatican Embassy. The US played rock music outside the Embassy until he decided to walk out and he was arrested.