The KLT (Kawai, Lewellen, Tye) formula famously maps a product of open string tree amplitudes to closed string tree amplitudes via a kinematic "KLT kernel" that depends on Mandelstam variables and alpha'. The low-energy expansion of this map gives the field theory double-copy of Yang-Mills theory to gravity via the leading order terms in the KLT kernel. In this talk, I will describe a novel systematic approach to define generalizations of the double-copy from the bottom-up and how it leads to a set of bootstrap equations for the double-copy kernel. I'll present new solutions to this bootstrap and demonstrate applications to the double-copy of YM as well as chiral perturbation theory with generic higher derivative operators. Finally I'll discuss potential implications for our understanding of the field theory landscape.