In a field theory it is possible to classify the contributions to the total energy as follows:
Energy intrinsic to the sources
This includes the 'bare' inertial mass of a source, disregarding field energy.
And it includes the self-energy of the source due to its own field.
But these two can be combined into a revised definition of the mass that makes no reference to a field. Hence this type of energy can be accommodated both in field theory and in DPI.
Energy due to the interaction between source and a given field
This case is often presented as a 'given' field acting on a nominally local source, but where that field is presumed to be sourced. Gravitational interaction not involving radiation is often symbolically expressed as due to a field (see here for example).
A field can be regarded as given if the response of the nominally local source does not produce a significant back-reaction on the sources of the 'given' field. The incoming given field can then be treated as decoupled from its own source. Generally this will be the case when the incoming field is the superposition of many sources whose motion can be characterized sufficiently for some calculation of their effect on a local source. This is the case for EM radiation from a star, characterized principally by its black-body temperature, say.
The supposedly de-coupled field provides a convenient though approximate summary of the effect of distant sources on a local source. But any such source-field interaction can, in principle, always be re-cast as direct particle interaction, though it may be mathematically inconvenient to do so. That is, the field is a useful but unnecessary construct.
Returning to the source-field illustration of gravity given above: Newton's original formulation was in terms of direct particle interaction. The field concept of gravity that allowed for its re-expression as source-field interaction came later with Poisson, and ultimately Einstein. In any case, the dynamics predicted by the two presentations of the theory are the same.
The question may arise: is a field decoupled if it is generated by distant sources on the backwards light-cone of the local source? More specifically, in QED, is single photon emission from a dipole pair of sources decoupled from its subsequent absorption by a dipole pair of sources? The short answer is no: the photon is not 'on-shell', and the interaction can be re-cast as direct interaction between dipoles. Such interaction is intrinsically time-symmetric and therefore reciprocal. In the absence of additional extraneous forces the two dipoles will always be coupled by exchange of 'virtual' photons.
Radiation into empty space destined never to be absorbed
The scope is restricted to electromagnetic radiation (including visible light), and gravitational radiation.
Both EM and gravity allow radiation in three dimensions away from a compact source (derived respectively, from one or more oscillating dipoles or quadrupoles. The development of such radiation in time can be illustrated by analogy with water waves, though these are restricted to two dimensions.
As discussed above, any alleged radiation destined to be absorbed can always be re-characterized as an interaction. In the parlance of field theory such radiation is not 'on-shell' and in this context is not true radiation.
It would appear that true radiation demands the existence of a field (technically: independent field degrees of freedom), and cannot therefore be accommodated by STF DPI (Schwarzschild, Tetrode and Fokker presentation of direct particle interaction). Certainly this was the position of Wheeler and Feynman (see history). They pointed out that STF DPI would remain viable if all radiation is destined to be absorbed, since then it would not strictly be true radiation, and then could be accommodated under the heading above (Energy due to the interaction between source and a given field).
The history page recounts the seeming failure of the effort by Wheeler and Feynman due to the fact that the present era universe was subsequently found to be mostly transparent to EM radiation. I.E. most radiation emitted now is destined never to be absorbed. This is the current mainstream view - though it turns out the mathematics of EM combined with cosmology allow for other outcomes.
It turns out however that radiation can be accommodated in a version of DPI that differs from that of Schwarzschild, Tetrode and Fokker, without the introduction of traditional field degrees of freedom. Though the starting point for this version of DPI is not much different that of STF, an outcome is the accommodation of two distinct `modes` whose equations of motion closely resemble those of charged matter and radiation. The perceived incompatibility between DPI and the radiation of energy `into empty space` goes away in this version of DPI because any such radiation is still hosted by particle degrees of freedom.
Absorption of un-sourced radiation
This possibility is allowed by the mathematics of field theory. We know of no examples, nor re-interpretation of observations consistent with this possibility. It is included here for completeness, and in case it eventually finds a home in a future development.
Field degrees of freedom that are forever uncoupled from matter
Field theory permits the field degrees of freedom - field 'modes' - that are not sourced, and destined never to interact with matter. In the classical theory this possibility is generally not exercised, in which case one could say these modes are empty. By contrast quantum theory, in particular quantum electrodynamics (QED), mandates these modes be occupied. Collectively they are referred to as the electromagnetic Zero Point Field (EM ZPF). Due to the presumption of a 4D space-time continuum the energy density of the EM ZPF is predicted to be infinite, which implies to some a pathology in QED.
Observations such as the Casimir Effect and the Lamb Shift were originally taken to be proof of the existence of the EM ZPF (by Bohr for example). However, according to the definition above wherein the ZPF is a constant non-interacting 'background', this evidence is at best indirect. Since the Casimir effect was first discovered and attributed to the ZPF Schwinger has shown it can be explained instead in terms of source-source interactions (consistent with DPI) without recourse to the EM ZPF.
Perhaps one could argue that the ZPF could be detected through its cosmological effect via General Relativity, On the largest scale GR reduces to the Friedman Equation which predicts the evolution of distances and therefore densities (of radiation, matter, and a few other drivers of the 'scale factor'). An infinite energy density would overwhelm all finite contributions of course, incapacitating the Friedman Equation. This would be contrary to the general view that the equation has some predictive power that is at least somewhat consistent with observation.
A complication is that Fermions have their own Zero Point Field, the energy density of which is negatively infinite. One can speculate that the total energy density of the two fields combined is finite, though there is nothing in QED that relates these two quantities.
In any case, the difficulties associated with these fields are absent from DPI, which seems to be a vote in its favor.