What are the end points in the action integral of field theory? In the mechanics of particles when we apply the principle of the least action the two end points are two spatial coordinates. Therefore, if we consider the variation of the action with respect to the variation of spatial variable, because these two end points are fixed, the term out of integral vanishes. 
But I wonder when we use the same variational approach on the Lagrangian density in field theory what are the end points? What we mean by "fixed" end points in this application of the principle? Are the fixed end points just the values of field on two events separated in spacetime? 
The path in the dynamics of a particle makes clear sense but "what is the path" in the variation of the Lagrangian density related to a field that we consider its two end points fixed?
 A: In field theory the values of the field at every point in space are independent degrees of freedom, just like the positions of different particles in a multi-particle system.  So, AFAIK to specify the initial and final configurations for an action integral you have to give the values of the field at every point in space at the initial and final times.
The "paths" that are being varied over are all the possible histories of the entire field, throughout space, that carry it from the initial to the final configuration.  When you find the path of least action, it specifies the behavior of the entire field over the time interval in consideration.
A: In particle mechanics you integrate along a path, which is bounded by points, but in field theory you integrate over a spacetime volume, so your boundary is a hypersurface, not just points.
For a typical quantum field theory process (at least the way it's formulated for calculations), there is some initial state consisting of noninteracting wavepackets, then there is a region of spacetime in which the wavepackets interact, then there is a final state which again consists of noninteracting wavepackets. In this case the volume of integration needs to contain the interaction. It's typical to pick an initial time well before the interaction and a final time well afterwards, and use those as the endpoints of the integral, i.e. you do
$$\int \mathrm{d}^4 r^\mu \to \int_{t_i}^{t_f} \mathrm{d}t \iiint \mathrm{d}^3 \vec{r}$$
