Whereby I assume a line segment of fixed length is uncontroversially finite. If we take a line segment that is increasing in length over time; does it really make sense to say the segment is finite in length? We may be able to say it was such and such a length at such and such a time in the past or in the future, but we would be unable to say what the length of the segment is now, for any utterance of its length immediately becomes incorrect as another interval of time passes.
One way of thinking infinity is that it measures an arbitrarily large set (or characterizes arbitrarily fast speeds, etc.)
The basic idea is that a given property is true regardless of how large something becomes or to what speed it accelerates.
Let's consider this object continuously increasing in size over time. It is clear it can become however large we want it given we are willing to wait some amount of time for the object to grow.
This seems to my mind to imply that any description of the size of this object will be correct at some point (given it is larger than the "initial" size of the object.) So, for instance, if we wish to assert of the size of this object will be 15m, and assuming the object began growing when it was smaller than this, at some point in time it will be 15m exactly.
As you suggest, if the object is enduring a continuous transformation of its size, while there is a precise moment when this property is true of the object, nevertheless its duration is infinitesimal, and it is already past before any indication is possible.
However it is of course possible to formulate rigorous functions to show the continuous variation in size, inferring a derivative rate of growth, which constitutes an adequate description at every moment in time. Functions are universal in this sense; they can capture transformations as variable maps.
As a quick commentary here, we are very close to the heart of science and mathematics: that is to say, the creation of functions, delicate sieves capable of capturing every point in a continuous transformation through abstraction, implicating the distribution of points in depth. Creating a function means plunging into chaos and extracting free variables to assemble new equations, isolating time or space, extruding the imperceptible lines of structural or genetic organization which articulate all the moments of growth and development.
Something more to think about: There actually are line segments which grow in length at all times and still never become longer then say, 2 meters. You can imagine a line segment, such that it's endpoints grow closer to any given two points in space, but just never quite reach them. So when imagining its limit shape, it actually is finite at all times...
As a second note, I feel you are concerned more with the principle of "now" then with the concept of finiteness of length.