Under which country's law is perjury across borders adjudicated?

The general rule of geographical jurisdiction in criminal cases is that conduct is actionable in both the place where it occurs and the place it causes harm. This general rule would apply, for example, when a statement is made under oath or under penalty of perjury, without knowing where it will end up being used.

However, testimony in a particular court, or a statement that states that it made under the penalty of perjury and cites to the law of a particular country is actionable only in that courts of the jurisdiction of the court in which the statement was made, or in the courts of the jurisdiction referenced in the document.

If a person submits testimony to a count in writing, this is most often done via an affidavit, which is generally sworn or affirmed to be the truth before a notary, magistrate, or other official. It can also, in some cases be done by signing a document which includes the words "under penalty of perjury" or some similar phrase, particularly if there is a law requiring such a document to be truthful.

If such written testimony or document is false in a material way, the country in which it was submitted to a court can, if it chooses to, prosecute this as perjury. (It may also be a contempt of court.) There would the the problem of enforcing any criminal judgement, which is always present when a person is charged with a crime by a country where the accused is not present. One possibility is to have the accused extradited from the country where the accused is present. This requires the cooperation of the source country, often via a treaty. Another possibility in some countries is to try the accused in absentia, and then arrest or extradite the accused later, if it becomes possible. This is more likely to be done if the country that wants to prosecute cannot obtain the accused's extradition.

In addition, when a person falsely swears to or affirms under penalty of per4jury a document, knowing that it will be sent to a court, or used for legal purposes, in another country, that person probably commits perjury in the country where the document was sworn to or affirmed, depending on that country's laws. That country could choose to prosecute the person locally, rather than extraditing the person.

Generally perjury must be over a serious issue, or neither county will choose to prosecute. It can be hard to prove. But that says nothing about which country has the right to prosecute, only about whether those rights will be acted on.

A few things:

1. perjury is perjury. It doesn't matter how you do it. What matters is that you have the obligation to tell the truth to the Court - in the eyes of that court. Simple example: I swear an affidavit when I'm in the US, that I know is going to be used in a UK Court. I lie in the affidavit. As far as the UK Court is concerned it is perjury.
2. There are international conventions whereby the courts of a country can "request" that the courts of another country "assist" with the taking of evidence from witnesses in that foreign country - let's say by letter. You'd have a duty in the foreign country to tell the truth. But we're not talking about that here.

Onto the if so:

3. Every court has a jurisdiction limited by geography.
4. Each court has power to punish people for contempt of court (arising from the perjury) within their jurisdiction
5. As a matter of public international law, foreign courts will not enforce (amongst other things) the penal laws of another country within their own country.
6. If what you've done is serious enough and an extradition treaty exists, you may find yourself extradited from the foreign country to be prosecuted in the originating court.
7. I'm not sure that you'll find statistics on instances such as these.

The lie probably tanked the (hypothetical) person's prospects of success who you thought you were helping in the first place. It's pretty hard to get one over on any court. Check out how your evidence would be tested.

By letter? No - telling lies is not perjury.

To commit perjury you must be under an oath or affirmation. In general, if a foreign jurisdiction is taking the affidavit then you would be subject to the relevant laws in both jurisdictions and either or both could take action against a false witness for perjury.

For example, s26(1) of the NSW Oaths Act 1900 says:

(1) Any oath declaration or affidavit required for the purpose of any court or tribunal or for the purpose of the registration of any instrument in this or any other State or Territory or the Commonwealth or for the purpose of any arbitration in this or any other State or Territory or the Commonwealth may be taken or made:

(a) in any place in this State before any justice of the peace for this State, and

(b) in any country or place out of this State before a notary public, or before any person having authority to administer an oath in that country or place, and

(c) in any country or place out of this State before a British Consular Officer or an Australian Consular Officer exercising his or her functions in that country or place.

Giving false testimony to that "person having authority to administer an oath in that country or place" would be an offence under the NSW Act and (presumably) "in that country or place."