Authorities in Barbados said they helped rescue several people from a flooded home, while school was cancelled in the nearby islands of St. Lucia, Dominica, Guadeloupe and Martinique.
Meteorologists said that up to 10 inches (25 centimeters) of rain fell in some parts of Martinique, Dominica and Barbados as they warned of flash floods and mudslides.
The U.S. National Hurricane Center also warned of heavy rains for St. Croix and eastern Puerto Rico.
Kirk had maximum sustained winds of 40 mph (65 kph) and was located about 270 miles (435 kilometers) south-southeast of Puerto Rico. It was moving west at 14 mph (22 kph).
The storm is expected to become a tropical depression by Friday night and degenerate into a low pressure trough by late Saturday.
Meanwhile, Hurricane Rosa weakened slightly off Mexico's Pacific Coast. The Category 3 storm had maximum sustained winds of 120 mph (195 kph) and was located 625 miles (1,005 kilometers) southwest of the southern tip of Baja California. It was moving west at 6 mph (9 kph).
It is expected to become a tropical storm by late Sunday as it heads northeast toward the Baja California peninsula. The National Hurricane Center said there were no coastal watches or warnings in effect, but swells generated by Rosa were expected to cause dangerous surf and rip current conditions along portions of southwestern Mexico, the peninsula and southern California.