* Shi'ite Houthis seek to tighten grip on north Yemen
* Western, Gulf states urge them to back peaceful change
* Yemen's instability poses risk to oil giant Saudi Arabia
SANAA, Aug 19 (Reuters) - Countries trying to save Yemen from falling apart at the hands of Islamist militants and separatist movements warned Shi'ite Muslim Houthis on Tuesday to stop trying to gain territory by force and engage in a political transition process.
Western and Gulf Arab countries are anxious to restore stability and security in Yemen, home to an al Qaeda wing that has sought to attack international airliners and targets in neighbouring Saudi Arabia, the world's top oil exporter.
The Houthis, who seek more power for the Zaydi Shi'ite Muslim sect in north Yemen, battled the central government in the last decade, but committed to peaceful transition after autocratic president Ali Abdullah Saleh was ousted in 2011.
Western and Gulf sponsors of democratisation in Yemen are keen to engage the Houthis because they have made big gains in popular support across north Yemen since the 2011 uprising and are able to bring large numbers of people onto the streets.
But they have again resorted to military force this year to consolidate their hold on the north, advance towards the capital Sanaa and fight rival Sunni factions, threatening the delicate transition of power in the turmoil-prone country.
On Monday the Houthis mobilised tens of thousands of supporters to protest against cuts in fuel subsidies after their leader, Abdul Malek al-Houthi, warned the government in a speech "against any aggression against our revolutionary Yemeni sons".
The Group of 10 Ambassadors, an informal body of Western and Gulf countries and international organisations involved in nudging along political reform in Yemen, wrote to the Houthi leadership on Tuesday voicing concern at recent statements it described as antagonistic and militaristic.
"SEEK CHANGE THROUGH POLITICAL MEANS"
"Any action aimed to incite or provoke unrest and violence is unacceptable and will be strongly condemned by the international community," said the statement, posted on websites of embassies of the group, which includes the United States, Britain, France, China, Saudi Arabia and the United Nations.
"We demand that you (Houthis) seek political means to express yourselves, not through force but through transparent, peaceful participation in Yemeni political life as a political party."
Monday's Houthi-organised mass rally in Sanaa posed yet another challenge to President Abd-Rabbu Mansour Hadi, a former Saleh deputy who has struggled to restore stability to U.S.-allied Yemen, the Arab world's poorest state.
The Houthis defend their territorial advance as simply an outgrowth of their growing popular support in those areas, but critics see an attempt to strengthen their position at a critical moment in the transition process.
The majority Sunni country is edging towards a federal system that would devolve more power to the regions. The national unity government formed after Saleh quit excludes Houthis, although Hadi has promised a more inclusive government after a new constitution is passed later this year.
April Longley Alley, senior Arabian Peninsula analyst at the International Crisis Group think-tank, said the ambassadors might have a limited amount of leverage to encourage the Houthis to make concessions.
"The Houthis do care about their international reputation and value the fact that they are now part of a national level political process with a seat at the bargaining table."
(Reporting by Mohammed Ghobari in Sanaa and Angus McDowall in Riyadh; Editing by Mark Heinrich)