I don't think there's a single simple analytical formula to find the intersection of a moving circle and a line segment. There's just too many elements in play. There are cases where the path of the circle doesn't intersect the line but the circle does (either tangent to the line or intersects in 2 places), or the circle's path passes through the infinite line but not the line segment, and so on. There are simply too many cases to run through to get a single formula.
However, there are a lot of tricks you can use. Circle-line intersections and circle-circle intersections can be very quick since they are analytical formulas. If you define your line segment as parametric in a certain way, your desired t values will be between 0 and 1 and everything else will be outside that range. Which tricks you use really depend on your desired approach.