* Mexican peso hit by trade tensions, Fitch downgrade
* S.Africa rand still pressured by row on c.bank mandate
By Susan Mathew
June 6 (Reuters) - Emerging market shares and currencies fell on Thursday on renewed growth and trade tensions after U.S. President Donald Trump said not enough progress was being made in talks with Mexico aimed at averting U.S. tariffs on Mexican goods.
Currencies were led lower against a weaker dollar by a 0.6% slip in the Mexican peso, which was also hit with a credit ratings downgrade for Mexico by Fitch. South Africa's rand meanwhile traded flat after touching eight month lows amid a dispute over changing the central bank's mandate.
Emerging market shares looked set to extend losses over a third session, down 0.2%, with most markets across Asia in the red a day after hopes of a cut in U.S. borrowing rates had provided some support to stocks.
After Trump said "not enough" progress was made on ways to curb illegal migration into the United States via the Mexican border at a meeting on Wednesday, investors will be watching for headlines when talks resume on Thursday in Washington.
Economists and analysts warn that the tariffs due to take effect on Monday will be negative for both economies and may tip Mexico, Latin America's No. 2 economy, into recession.
"Until we see some more encouraging measures I think EM assets are going to trade on the back foot," said Chris Turner, ING's global head of strategy and head of EMEA and LATAM research.
Mexico's peso fell up to 1.3% and was last trading down 0.6% at 19.7023 per dollar.
Late on Wednesday rating agency Fitch cut Mexico's sovereign debt rating to BBB, nearing junk status, citing risks from heavily indebted state oil firm Pemex and from trade tensions, while Moody's lowered its outlook to negative.
This was followed by Trump's comments.
The peso has fallen about 2.5% since Trump's surprise threat last week to levy tariffs on all Mexican imports at an initial 5% and rising up to 25%.
South Africa's rand traded marginally lower. A dispute between the ruling African National Congress and government over the mandate of the central bank, spurred by data showing the worst economic contraction in a decade, pressured the currency.
"We don't see it as a story with longevity," ING's Turner said. "We don't think there will be a change in the central bank's mandate. There may be some clarifications from the government on the issue and that should allow the story to blow over."
India's rupee gained after the central bank cut interest rates by 25 basis points as widely expected and changed its monetary policy stance to "accommodative" after the economy grew at its slowest pace in over four years in January-March.
Russia's rouble climbed 0.2%, with higher crude prices aiding the currency, while stocks rose 0.9%.
Hawkish Russian central bank governor Elvira Nabiullina flagged risks from policy easing to inflation on Thursday and said the key interest rate was now above the neutral range. The bank has already said it is ready to consider cutting rates in the near future, she noted.
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