For most flutes you can get all the readings you need using the following two Starrett Telescoping gages with 12" handles:
579A-12 Telescoping Gage: 5/16 - 1/2"/8-13mm Range, 12"/300mm Handle Length
579B-12 Telescoping Gage: 1/2 - 3/4"/13-19mm Range, 12"/300mm Handle Length
I actually have the next larger one too (579C-12) but I only need it for flutes in lower keys than D.
I take measurements at regular intervals along the bore (either every 10 mm or every 5 mm, depending on how detailed I want to be).
The reference point for my measurements is the start of the bore (head end of the upper body section), and this is where I insert the gage. The shaft of the gage can be marked with the relevant distances you want to insert.
The section of the bore from that point to the embouchure is generally cylindrical (ignoring for now the steps due to tuning slide), so it can be simply measured at each end. Its length varies depending on the tuning slide extension, so I keep that part of the data separate and only really use it when making a new head for a flute, perhaps with some head length correction for playing at modern pitch with less tuning slide extension (and hence less of a tuning slide cavity in the bore).
You can profile each body section and foot separately (inserting the gage from the wide end) and then measure lengths of the sections and do the calculations needed to convert your measurements to distances from the reference point mentioned above. Then you can graph the bore as a whole. But make sure you remember to take into account the socket depths and tenon lengths correctly, and verify that the tenons are the same length as the sockets they fit into! Sometimes they are not!
Most antique wooden flutes have experienced movement due to shrinkage, and this results in the bore being oval in places. You can test for, and account for, this by taking widest and narrowest measurements at each point and by averaging the two. Acoustically, it is the cross sectional area of the bore that is most important. This is one advantage of using these telescoping probes vs pre-machined cylindrical probes. A cylindrical probe can only measure the narrowest dimension. Of course, you can’t quite do this perfectly at the location of the tone holes.
By taking measurements in this way and using them to carefully construct reamers you can make quite precise copies of antique flutes, which play like the originals. Not in their original state, of course, but in their current state, which is sometimes what you want.
Rod Cameron used similar techniques to this, and I learned quite a bit from his posts years ago, but he also made a device for scanning the interior of a bore without the need for metal on wood contact, which in theory risks damaging the flute. In practice, though, I’ve never noticed any damage to the bore from using the kind of gages linked above. They have very smooth, rounded ends, and are spring-loaded with fairly weak springs. You lock them off before removing them, but so long as the bore is not actually chambered (containing a local maximum between two narrow points) removing them to the wide end is simple. If it is chambered (and not just at the point where sections meet) you can’t get accurate measurements of the chamber width using these gages because you lose the measurement in order to extract the gage.