Is it permissible to catch frogs and, after slaughtering, export them to lands where they are eaten?

This is a multifaceted issue since it discusses the rulings of catching, slaughtering, and exporting frogs to be eaten. The issue of exporting frogs is built upon the issue of slaughtering them, and the matters return to the ruling of killing them. There are hadiths which forbid killing frogs. One of them is the hadith narrated by ‘Abd Allah ibn ‘Uthman who said that the Prophet forbade killing frogs [Ahmed, Abu Dawud, al-Nasa’i, Ibn Majah; al-Hakim declared it authentic and al-Dhahabi remained silent].

Abu Huraira (may Allah be pleased with him) reported that the Prophet forbade killing shrikes, frogs, ants and hoopoes [Ibn Majah].
‘Abd Allah ibn ‘Omar (may Allah be pleased with them both) said that

The Prophet forbade killing frogs and said: “Its croaking is tasbeeh (praising God)." [al-Tabarani in al-Saghir and al-Awsat]

In his Sunnan, al-Bayhaqi included the hadith narrated by Ibn 'Amr (may Allah be pleased with them both):
Do not kill frogs; indeed, its croaking is tasbeeh.
al-Bayhaqi said that its chain of transmission is authentic.

Because of this, the Hanafis, Shafi‘is, Hanbalis, Zahiris and many other scholars consider it prohibited to eat frogs due to the juristic rule that everything which has been forbidden to kill is [also] forbidden to eat, since if it were permissible to eat it would have been permissible to kill.

Some other scholars permit consuming frogs, based upon the generality of the words of Allah the Almighty:
It is permitted for you to catch and eat seafood [5:96]

and due to the generality of the well known hadith: “Its water is purifying and its dead are lawful." This opinion is reported from Malik, Ibn Abi Laila, al-Sha‘bi, and al-Thawri in one of his two opinions. In addition to this, they considered the hadiths about killing frogs to be weak.

We side with the opinion held by the majority scholars who prohibit eating frogs based upon their prohibition of killing it, which is itself based upon the ‘People of Hadith’ considering the hadith concerning the subject to be fair.

Based on the above and in reference to the question: it is impermissible for you to hunt, slaughter, and export frogs.
And Allah knows best.