Can he slaughter an animal with the intention of giving the meat in charity in his own country or can he choose a poorer country?

If you have relatives and neighbours in your country who are poor, then giving voluntary charity to them is better, and starting with them is better. The religious texts give precedence to them and encourage us to start with them.

Allah, may He be exalted, says (interpretation of the meaning):

{Say: Whatever good you spend should be for parents, kindred, orphans and those in need, and for wayfarers, and whatever good you do, Allah is aware of it}[Al-Baqarah 2:215].

It was narrated that Abu Hurayrah said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “A dinar that you spend for the sake of Allah, a dinar that you spend to free a slave, a dinar that you give in charity to a needy person and a dinar that you spend on your family – the greatest of them in reward is the one that you spend on your family.” Narrated by Muslim (995).

It was narrated that Abu Hurayrah said: The Prophet (blessings and peace of Allah be upon him) instructed us to give charity, and a man said: O Messenger of Allah, I have a dinar. He said: “Give it in charity to yourself.” He said: I have another. He said: “Give it in charity to your child.” He said: I have another. He said: “Give it in charity to your wife – or he said: your spouse.” He said: I have another. He said: “Give it in charity to your servant.” He said: I have another. He said: “You know best [to whom to give it].” Narrated by Abu Dawud (1691).

Al-Nawawi (may Allah have mercy on him) said: The ummah is unanimously agreed that charity given to relatives is better than that given to outsiders, the hadiths which speak of this matter are many and well-known. Our companions said: With regard to it being recommended to give voluntary charity to relatives and give them precedence over outsiders, it makes no difference whether the relative is one on whom you are obliged to spend or not. Al-Baghawi said: Giving it to a relative on whom you are obliged to spend is better than giving it to an outsider."(Al-Majmu‘  6/238).

Secondly:

If the reasons to give precedence to those mentioned above are not there, then undoubtedly finding another country that is poorer to slaughter an animal there, with the intention of giving its meat in charity, is more appropriate. Hence we see that the verse of zakah gives precedence to those in the greatest need and starts with them. Allah, may He be exalted, says (interpretation of the meaning):

{Zakah [alms] is only for the poor and those in need; those who work to collect it; those whose hearts are to be won over; for the freeing of slaves; for debtors; for the cause of Allah; and for stranded wayfarers. [Thus it is] ordained by Allah, and Allah is All-Knowing, Most Wise}[Al-Tawbah 9:60].

Thus the verse begins with the poor, who are the most in need.

Therefore a number of scholars allowed sending obligatory zakah to a country that is poorer and in greater need, so it is more appropriate that sending voluntary charity should be permissible.

Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said: There is no shar‘i evidence for defining a limit on the distance beyond which zakah cannot be sent and saying that it is the distance at which shortening the prayers becomes permissible. It is permissible to transfer zakah, and whatever comes under the same ruling, for a shar‘i reason."(Taqrib Fatawa Ibn Taymiyah  3/514).

It says in a fatwa of the Permanent Committee: “The basic principle regarding zakah is that it should be given to the poor in the city where the wealth is, because of the hadith quoted, but if there is a need to send it elsewhere, such as if the poor people in the country to which it is sent are in greater need, or they are relatives of the one who wants to give it, in addition to being poor, or for some similar reason, it is permissible to transfer it."(Fatawa al-Lajnah al-Da’imah 10/9).

Shaykh al-Sa‘di (may Allah have mercy on him) said: The giver should seek out needy people to give his charity to, and he should not give it to a needy person when there is someone in greater need than him."(Tafsir al-Sa‘di  p. 116).

Based on that, if there are people in the country in which you want to slaughter the animal with the intention of giving its meat in charity who are in greater need than those in your city, because of calamities, war and the like, then it is better to slaughter the animal there, not because the price is cheaper, but in order to meet the need of those who have been stricken by poverty or because some disaster has destroyed the wealth of the Muslims. In the case of voluntary charity, the matter is more flexible.

And Allah knows best.