The manipulation of droplets with sizes on the millimetre scale and below has attracted considerable attention over the past few decades for applications in microfluidics, biology, and chemistry. In this paper, we report the response of an oil droplet floating in an aqueous solution to local laser heating. Depending on the laser power, distinct dynamic transitions of the shape and motion of the droplet are observed, namely, breathing, crawling, budding, and splitting. We found that the selection of the dynamic modes is determined by dynamic instabilities due to the interplay between the convection flows and capillary effects. Our findings can be useful for constructing microfluidic devices to control the motion and shape of a small droplet by simply altering the laser power, and for understanding thermal convective systems with fully soft boundaries.