The paper considers a problem of exact observability for nonstationary linear hyperbolic systems with nonstationary interior observations. Three types of scanning observations are considered and the existence of measurement curves and set-valued maps for sensors that ensure exact observability is established. The proposed method is based on a priori energy estimates of instantaneous type for solutions. It allows to obtain required measurement curves that are continuous in a fixed interval or measurement maps that are lower semi-continuous in the interval with respect to Lebesgue measure.