Norway’s proposal to sell off $35 billion in oil and natural gas stocks brings sudden and unparalleled heft to a once-grassroots movement to enlist investors in the fight against climate change.

The Nordic nation’s $1 trillion sovereign wealth fund said Thursday that it’s considering unloading its shares of Exxon Mobil Corp., Royal Dutch Shell Plc and other oil giants to diversify its holdings and guard against drops in crude prices. European oil stocks fell.