Miami, Florida (CNN) -- Tropical Storm Alex continued to weaken Thursday as it slogged across Mexico, with top sustained winds decreasing to near 40 mph with higher gusts, the National Hurricane Center said.
Alex was expected to drop an additional 3 to 6 inches of rain across portions of northern and central Mexico, the center said in its 5 p.m. ET update.
Total accumulation could reach 20 inches in higher areas of northeastern Mexico, it said. "These rains could cause life-threatening flash floods and mudslides, especially in mountainous terrain," it said.
Another 2 to 4 inches of rain were expected to fall on parts of southern Texas, bringing the storm's total deposit to about 12 inches and raising the specter of life-threatening flash floods there, too, it said.
At 5 p.m. ET, Alex was centered about 95 miles (155 kilometers) east-northeast of Zacatecas, Mexico, moving westward at 13 mph (20 kph), it said.
Despite continuing heavy rain in portions of southern Texas and northern Mexico, Alex was expected to dissipate Thursday night over Mexico's high terrain, it said.
The season's first named storm was downgraded earlier Thursday from hurricane status after making landfall Wednesday night as a Category 2 hurricane in the Mexican municipality of Soto la Marina, the hurricane center said.
Mexico's weather service said the storm's center made landfall with 96 mph winds in San Fernando in the Mexican state of Tamaulipas, 75 miles south of the border, according to meteorologist Martin Tellez.
All tropical storm watches and warnings have been discontinued, but a tornado watch remained for south Texas until 8 p.m. CT (9 p.m. ET).
Even though Alex was centered more than 600 miles from the massive BP oil disaster, officials said the storm could affect cleanup efforts for days because it stirred up heavy seas across the Gulf of Mexico.
Alex had already disrupted the use of containment booms meant to limit the amount of oil reaching shore, Coast Guard Rear Adm. Paul Zukunft told reporters Wednesday.
In addition, high seas led more than 500 oil-skimming ships to return to shore and officials suspended efforts to burn oil on the surface and break it down with dispersants, Zukunft said. Also delayed was a move to position a third ship to collect oil at the spill site.
Skimming and burning of oil may not resume before Saturday or Sunday, CNN meteorologist Chad Myers said.