The field of space communications is the realm of communication technologies where diffraction and atmospheric effects, both of which contribute to loss and noise, become overriding. The pertinent questions here are how and at which rate information (secret keys) can be securely transferred (shared) among users under such supposedly severe circumstances. In the present work we study continuous-variable (CV) quantum key distribution (QKD) in a measurement-device-independent (MDI) configuration over free-space optical (FSO) links. We assess the turbulence regime and provide a composable finite-size key rate analysis of the protocol for FSO links. We study both short-range, horizontal communication links as well as slant paths to, e.g., high-altitude platform station (HAPS) systems.