It is useful to divide the problem up.
How do the fields vary?
First understand that each configuration of currents (be they classical or quantum generates a particular contribution to the magnetic field. Call such a contribution a "source field".
In your case each magnet generates a source field that is roughly an extended dipole (and the lines do run as loops: the figure has simply neglects to show the lines inside the magnets). These sources field move and re-orient as you move the magnet but they are unaffected by other magnets (up to a point, anyway).
The magnetic field is the sum of all source fields.1
Magnetic fields are vectors2 so the word "sum" means "vector sum".
How do the field lines look?
Field lines are a visualization of fields.
They are drawn so that:
- At every point on a drawn line the field is tangent to the line.
- In any given region of space the strength of the electric field is proportional to the local density of field lines3
This representation can be quite lossy if you use too few lines (and most figures use too few to show much detail). Drawing a representative set of field lines from an arbitrary field is a bit of an art.
But if you have lost information in draw the lines, then you can't properly reconstruct the sum staring with only the line drawings and can't "add" the field line drawing directly.
How to construct a visualization of the sum?
The proper way to construct a detailed and accurate field-line maps for interactions between the magnets is to start by measuring or approximating the source fields,4 then add them in vector arithmetic, then draw a suitable set of field lines of the sum.
1 Usually we restrict the sum to strong, nearby sources for practical reasons, but in principle you should be summing over the retarded contribution of all sources in the historical light-cone of your event of interest. Luckily distant contributions tend to be small enough that we really don't care.
2 In the introductory treatment, anyway. A move complete treatment makes them a skew-symmetric portion of the electromagnetic field tensor, but the ideas here-in can be extended to that treatment in a straightforward way.
3 I believe that to get the density bit quantitatively right you actually have to draw your lines in three-dimensions, but we just fudge it.
4 With a sufficiently dense drawing of field line you can make a good approximation of the field from the line-representation. Thus the weasel words about not being able to add field-line drawing directly.