This answer tells us that embassy soil is inviolable, as per Vienna Convention on diplomatic Relations 1961, art. 22. That means the host country ('s law enforcement) cannot enter the premises nor do anything inside.
I assume that applies only without the consent of the head of the diplomatic mission, but is that right? My understanding of the arrest of Julian Assange in UK by bobbies given access by the Ecuador diplomats seems to go that way.
What about the mission country's law enforcement? In the Agent 47 movie, US Marines make an arrest just in front of the building of the US embassy in Germany, so that seems like the mission country does have law enforcement power within the embassy grounds, but some other source than that not-so-realistic movie would be appreciated.
Assuming point 2 is correct, and a suspect is arrested, what happens next? If the suspect were to be moved to the mission country's sovereign soil (which the embassy is not), would that country's laws allow them to be prosecuted, or would the court consider the act they committed to have been done in foreign sovereign territory ?
Still assuming point 2 is correct, what would be the conditions for moving the suspect from the embassy (within the host country) to the mission country? can the host country forbid it?
Finally, what happens if, say, a hostage-taking attack prevents anyone from the embassy to communicate the permission given in point 1, and the law enforcement capacities from point 2 (from the mission country) are not capable of responding (because they are incapacitated for example)? The most extreme legal response would be for the host country to revoke diplomatic credentials for the diplomatic mission, and to invade it with its own law enforcement after a given period of time. Is there another way? For example, can a mission-country diplomat (the minister for foreign affairs for example) provide a point-1 authorization from abroad, without consent from the ambassador? Can the mission country send troops to the embassy through the host country? Can the host country "lend" troops to the mission country's command?