* Colombian peso drops 1.4% as oil prices slide * Chilean constitution drafting process in focus (Updates prices after Fed minutes,) By Ambar Warrick and Susan Mathew July 7 (Reuters) - Latin American currencies cut a chunk of their session losses on Wednesday after minutes of the Federal Reserve"s last meeting took the dollar a leg lower, while sliding oil prices pushed Colombia"s peso to two-month lows. Brazil"s real was down 0.4%, having lost up to 1.6% earlier on Wednesday to hit a one-month trough, while Mexico"s peso reversed its losses entirely. Rising copper prices helped Chile"s peso recover after losing as much as 1.3%. It was last trading down 0.5%. Federal Reserve officials last month felt that substantial further progress on the economic recovery "was generally seen as not having yet been met," but as a whole felt it needed to be prepared to act if those risks materialize. "It wasn"t a big change. I don"t think there"s any big takeaways from this other than they are going to begin talking about tapering at the next meeting officially, which is not unexpected," said Andrew Richman, senior fixed income strategist at Sterling Capital Management in Jupiter, Florida. Emerging market currencies had taken a hit after the Fed"s last meeting when it sounded surprisingly hawkish. Fears of a sooner-than-expected tapering brought back memories of a dive in 2013, while potentially higher U.S. interest rates narrow the gap between interest rates which make EM currencies attractive. Colombia"s peso dropped 1.4% to 3,828 per dollar. Oil prices slid as investors feared this week"s collapse in OPEC+ talks could mean more supply, not less, is on the way. Anti-government protests, the coronavirus pandemic, and a second credit rating downgrade have knocked the oil exporting Andean currency down almost 11% this year, compared to an over 40% rise in crude prices. In Chile, all eyes are now on the process for drafting a new constitution. Delegates on Sunday chose a woman from Chile"s majority indigenous Mapuche people to lead them in the process. Peru"s sol extended losses to a fifth straight session as thousands of Peruvians took to the streets on Tuesday to protest uncertainty over the result of the presidential election a month ago. Sources said socialist Pedro Castillo, expected to be confirmed as Peru"s next president, is drawing up a "hybrid" Cabinet of technocrats and political allies, with a moderate in line to run the economy ministry. He expects mining firms to accept "prudent" tax changes, a top adviser told Reuters Most Latam stock markets rose, with Brazil"s Bovespa pulling away from two-month lows. Key Latin American stock indexes and currencies: Stock indexes Latest Daily % change MSCI Emerging Markets 1340.19 -0.47 MSCI LatAm 2540.62 0.65 Brazil Bovespa 127051.43 1.56 Mexico IPC 50066.32 0.03 Chile IPSA 4174.43 0.21 Argentina MerVal 63014.96 -0.862 Colombia COLCAP 1286.56 1.14 Currencies Latest Daily % change Brazil real 5.2298 -0.38 Mexico peso 19.9485 0.31 Chile peso 750 -0.47 Colombia peso 3828 -1.37 Peru sol 3.9495 -0.21 Argentina peso 95.9400 -0.02 (interbank) (Reporting by Ambar Warrick; Editing by Jonathan Oatis and Andrea Ricci) Our Standards: The Thomson Reuters Trust Principles.