* Sterling wallows below overnight lows

* Dollar gains vs yen, pushes toward last week's highs

* Market awaits Fed minutes, Yellen speech this week

* Kiwi, Aussie tumble on dovish RBNZ

TOKYO, Oct 11 (Reuters) - The dollar firmed on Tuesday, while the beleaguered sterling wallowed near recent lows on lingering fears about the impact on Britain from exiting the European Union and the kiwi tumbled on dovish comments from a New Zealand central bank official.

Japanese, Canadian and some U.S. markets were closed on Monday for holidays.

The dollar index, which tracks the greenback against a basket of six major rivals, added 0.1 percent to 97.024.

"The U.S. dollar's grind higher continues, clearly linked to changing Fed rate hike expectations," said Sue Trinh, head of Asia FX strategy at Royal Bank of Canada in Hong Kong.

"A little over three weeks ago, we had expectations of a December rate hike at under 50 percent, and now we're pushing towards 70 percent," she said. "So the onus is on the data now, to continue printing firmly."

Market participants were pricing in around a 70 percent chance that the Fed will raise rates in December, according to CME Group's FedWatch program.

Investors awaited Wednesday's release of minutes of the Federal Reserve Open Market Committee's September meeting for clues as to how close the Fed is to hiking interest rates.

Speaking to reporters after a speech in Sydney, Chicago Fed President Charles Evans said on Tuesday that he "could be fine" with the Fed raising rates in December, but he wanted to see how the economy and inflation progressed before deciding.

Evans does not have a vote this year on Fed policy but participates fully in deliberations and will become a voting member in 2017.

"At the end of the day, it's a dollar strength story, and after that, it's currency positioning. That's really the driving factor," said Bart Wakabayashi, head of Hong Kong FX sales at State Street Global Markets.

Sterling, meanwhile, was down 0.3 percent against the dollar at $1.2324, below its overnight low of $1.2345, in the wake of Friday's "flash crash" that sent it hurtling to its lowest levels in 31 years.

"Economic data does not help very much to explain sterling's movement. The fact of the matter is that the UK has fared well in the three and half months since the referendum," wrote Marc Chandler, global head of currency strategy at Brown Brothers Harriman.

"The depreciation of sterling will have an impact on the UK current account balance," he said. "It will be reduced, but do not be surprised if it comes from reduced volume of imports as much as an increase in value of exports."