The USA, Canada, Britain, France, Australia, Russia have all had Islamist terrorist attacks on their own soil. Many of these countries have also had people joining Islamist groups overseas such as the Taliban and ISIS.
Japan's terrorist attacks have largely been by non-Muslims, but Hitoshi Igarashi, the Japanese translator of "The Satanic Verses" was murdered (although the perpetrator and motive isn't publicly known), and ISIS announced this year the killing of two Japanese hostages, claiming that they were in response to Japan's foreign policy. (I'm rather suspicious that they had already been murdered by ISIS prior to any announcements)
I haven't heard about any problems with political violence by Muslims in Mongolia. When searching about terrorism in Mongolia, the only information I found was an incident of eco-terrorism, plus news related to inner Mongolia in China.
One idea favoured by some on how to prevent political violence by Muslims is not to allow Muslims into a country, such as in the blog post ISLAMIC TERRORISM: WHY THERE IS NONE IN JAPAN. However, this isn't the case with Mongolia. About 3% of Mongolia is Muslim, mainly Kazakhs in the part of the country close to Kazakhstan.
Another idea on how to avoid political violence by Muslims is not to be involved militarily in Muslim countries. But while Mongolia has tried to play a neutral foreign policy, it has been involved in peace-keeping in Muslim countries, such as Iraq.
Has Mongolia been doing something right that other countries haven't done, and if so, what is it?