They also seek to establish a task force study of the consequences of global warming on poor economies, and called on Exxon to reduce greenhouse gas emission at its own operations.
Exxon is "profiting in the short term from investments and decisions made many years ago by focusing on the narrow path that ignores the rapidly shifting energy landscape around the world, including developing nations," said Neva Rockefeller Goodwin, a great granddaughter of John D. Rockefeller.
The family members, who describe themselves as the company's longest continuous shareholders, said they are concerned that the Irving, Texas-based company is too focused on short-term gains from soaring oil prices and should do more to invest in cleaner technology for the future.
"They are fighting the last war and they're not seeing they're facing a new war," said Peter O'Neill, who heads the Rockefeller Family committee dealing with Exxon Mobil and is the great-great-grandson of John D. Rockefeller.
He said he had the support of more than 80 percent of family members over the age of 21. Family representatives said it was a significant holding for the Rockefellers but that they were not sure how much of the company they actually own collectively.
Exxon Mobil was formed by the combination of two offspring of John D. Rockefeller's Standard Oil Trust. It is now the world's largest publicly traded oil company.
Members of the family said they have sponsored four proxy resolutions this year that raised concerns about the company's leadership under Chairman and Chief Executive Rex Tillerson. They also said they have spent years behind the scenes prodding the company to change its approach to the oil business.
'Trying to keep a giant ... from falling'
The family and its allies decided to take their case public, they said, because they believe future energy will come from sources other than oil and natural gas, and say the company needs to move more quickly into sustainable technology to secure its long-term viability.
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"We all know the saying: The bigger they are, the harder they fall," said Connecticut State Treasurer Denise Nappier, who oversees a pension fund that holds $300 million in Exxon Mobil stock - its largest single equity investment. She spoke at a press conference alongside the Rockefellers.
"We are trying to keep a giant - and it truly is a giant in the oil and gas industry - from falling," Nappier said.
Goodwin called on Exxon to reconnect with the forward-looking vision of her great grandfather.
"Kerosene was the alternative energy of its day when he realized it could replace whale oil," she said. "Part of John D. Rockefeller's genius was in recognizing early the need and opportunity for a transition to a better, cheaper and cleaner fuel."
Huge profit expected
The calls for reform came one day before Exxon Mobil was expected to report first-quarter earnings of more than $11 billion, according to according to a survey by Thomson Financial. Thanks to rapidly rising oil prices, that is considerably more than the company earned a year earlier, and could even top Exxon Mobil's own record for the biggest quarterly profit in U.S. history.
The company's board is recommending shareholders vote against a proposal to split the role of chairman and CEO. In a recent proxy statement filed with the Securities and Exchange Commission, the board said "that the most effective leadership structure for Exxon Mobil Corporation at the present time is for Mr. Tillerson to serve as both Chairman and CEO."
Exxon Mobil spokesman Gantt Walton said the company has met with members of the Rockefeller family on multiple occasions and "respects the rights of all shareholders to make their views known," but that it does not comment on details of meetings with shareholders.
The stock is up more than 63 percent since Tillerson became CEO on Jan. 1, 2006, compared with a gain of 11.4 percent for the broad S&P 500 index over the same period.
The Associated Press and Reuters contributed to this report.
© 2008 MSNBC