We investigate color-kinematics duality for gauge-theory amplitudes produced by the pure nonabelian Yang-Mills action deformed by higher-dimension operators. For the operator denoted by F^3, the product of three field strengths, the existence of color-kinematic dual representations follows from string-theory monodromy relations. We provide explicit dual representations, and show how the double-copy construction of gravity amplitudes based on them is consistent with the Kawai-Lewellen-Tye relations. It leads to the amplitudes produced by Einstein gravity coupled to a dilaton field phi, and deformed by operators of the form phi R^2 and R^3. For operators with higher dimensions than F^3, such as F^4-type operators appearing at the next order in the low-energy expansion of bosonic and superstring theory, the situation is more complex. The color structure of some of the F^4 operators is incompatible with a simple color-kinematics duality based on structure constants f^{abc}, but even the color-compatible F^4 operators do not admit the duality. In contrast, the next term in the alpha-prime expansion of the superstring effective action a particular linear combination of D^2 F^4 and F^5-type operators does admit the duality, at least for amplitudes with up to six external gluons. Show Partial Abstract