In one aspect of the teachings presented herein, an autonomously-navigating mobile robot moves to a designated location and detects the presence and relative positioning of a compatible wheeled cart, based on recognizing one or more characteristic physical features of the cart from sensor readings taken by the robot. The robot moves into a position of gross alignment with the cart, based on the detected relative position of the cart, which means that the robot accommodates carts that are not properly positioned or oriented for pickup. The robot takes additional sensor readings referenced to the same or additional characteristic features of the cart and moves into a position of fine alignment with the cart, whereupon it latches to the cart and transports it via rolling conveyance to a designated destination location. The cart preferably includes a normally-on brake that is automatically disengaged by virtue of the robot latching to it.