The integral of functions defined on different meshes

Hello, I have a small question if you could help me it would be great.
I work on a problem which depends on time, and I do mesh adaptation
according to a posteriori error indicators.

In the variational formumation of my time dependent problem, there is a
term which corresponds to the integral of the value of my unknown at the
previous time step times the basis functions. When the mesh is adapted,
I am wondering how this integral is computed since there is a product of
a finite element function supported by the previous mesh (the unknown at
the previous time step) with the basis functions on the new mesh.
I guess there must be some interpolation process involved in this
calculation, along with quadrature formulas. I found something in the
FreeFem++ Documentation (Release 4.8) in section 3.3.11 entitled “A Fast
Finite Element Interpolator” and so my question is to make sure that
this is the actual process involved in the calculation of the integral I
described above, and if not to know how this integral is computed?

Thank you!

When FreeFem++ set a finite element function u it all store a pointer to the courant meshes.
So after it you change the mesh and the do no change on the finite element function u in interpoled
at quadrature points of new meshes, and this process is apply every where.