  We study exact solutions of the infinite derivative gravity with null
radiation which belong to the class of almost universal Weyl type III/N Kundt
spacetimes. This class is defined by the property that all rank-2 tensors
${B_{ab}}$ constructed from the Riemann tensor and its covariant derivatives
have traceless part of type N of the form $\mathcal{B}(\square)S_{ab}$ and the
trace part constantly proportional to the metric. Here, $\mathcal{B}(\square)$
is an analytic operator and $S_{ab}$ is the traceless Ricci tensor. We show
that the convoluted field equations reduce to a single non-local but linear
equation, which contains only the Laplace operator $\triangle$ on 2-dimensional
spaces of constant curvature. Such a non-local linear equation is always
exactly solvable by eigenfunction expansion or using the heat kernel method for
the non-local form-factor $\exp(-\ell^2\triangle)$ (with $\ell$ being the
length scale of non-locality) as we demonstrate on several examples. We find
the non-local analogues of the Aichelburg--Sexl and the Hotta--Tanaka
solutions, which describe gravitational waves generated by null sources
propagating in Minkowski, de Sitter, and anti-de Sitter spacetimes. They reduce
to the solutions of the local theory far from the sources or in the local
limit, ${\ell\to0}$. In the limit ${\ell\to\infty}$, they become conformally
flat. We also discuss possible hints suggesting that the non-local solutions
are regular at the locations of the sources in contrast to the local solutions;
all curvature components in the natural null frame are finite and specifically
the Weyl components vanish.
