Does direct interparticle action imply advanced inertial forces? In his Nobel lecture Richard Feynman states that by varying the Schwarzschild-Tetrode-Fokker direct interparticle action
$$A=-\sum_i m_i\int\big(\mathbf{\dot X_i}\cdot\mathbf{\dot X_i}\big)^{1/2}d\alpha_i+\frac{1}{2}\sum_{i\ne j}e_ie_j\iint\delta(I_{ij}^2)\ \mathbf{\dot X_i}\cdot\mathbf{\dot X_j}\ d\alpha_i\ d\alpha_j\tag{1}$$
where
$$I_{ij}^2=\big[\mathbf{X_i}(\alpha_i)-\mathbf{X_j}(\alpha_j)\big]\cdot\big[\mathbf{X_i}(\alpha_i)-\mathbf{X_j}(\alpha_j)\big]$$
one can reproduce classical electrodynamics without using the concept of the electromagnetic field.
In their paper Classical Electrodynamics in Terms of Direct Interparticle Action John Wheeler and Richard Feynman showed in the section Action and Reaction on pages 429-430 that the energy-momentum transferred by retarded forces from particle $i$ to particle $j$ along a null worldline connecting them is equal and opposite to the energy-momentum transferred by advanced forces from particle $j$ back to particle $i$ along the same null worldline. Therefore they had discovered a Lorentz covariant generalization of Newton's principle of action and reaction.
Following the Feynman Lectures vol.1 ch. 28 section 28-2 I imagine two stationary particles with charges $e_1$ and $e_2$ separated by a large distance $r$ so that only the radiative electromagnetic forces, which decay as $1/r$, are relevant. 
Let us suppose that I apply a contact force to particle $1$ at time $t$ that gives it an acceleration $\mathbf{a}(t)$ perpendicular to the line joining the two particles.
The retarded electromagnetic force received by particle $2$ at time $t+r/c$, having been emitted by particle $1$ at time $t$, is given by
$$\mathbf{F^{21}}(t+r/c)=\frac{-e_1 e_2}{4\pi\epsilon_0 c^2r}\mathbf{a}(t)\tag{2}$$
For simplicity let us suppose that the mass of particle $2$ is very large so that it hardly accelerates at all and therefore does not produce a retarded force back on particle $1$.
However if the direct interparticle action $(1)$ is a correct description of Nature then there should be an advanced reaction force back on particle $1$ at time $t$, emitted by particle $2$ at time $t+r/c$, given by
$$\mathbf{F^{12}}(t)=\frac{e_1 e_2}{4\pi\epsilon_0 c^2r}\mathbf{a}(t)\tag{2}$$
As this back reaction force is proportional to the acceleration then it will manifest itself as an apparent change in the inertia of particle $1$, $\Delta m_1$, given by
$$\Delta m_1=\frac{e_1 e_2}{4\pi\epsilon_0 c^2r}\tag{3}$$
This apparent change in inertia of particle $1$ should be detectable. Has any such effect been measured? 
 A: Feynman et. al. are in Ref. 1 deriving  a relativistic generalization of Newton's 3rd law in a closed system of point charges. However OP's momentary contact force to particle 1 constitutes an external force unless it is explained via say a 3rd point charge of the system, so OP's 2-particle scenario is at best an incomplete description.
Below we sketch a derivation of the interesting non-local direct interparticle action used by Wheeler & Feynman, which goes back to Schwarzschild, Tetrode & Fokker.

*

*We start we with the Maxwell Lagrangian density$^1$
$$ \begin{align}{\cal L}_0
~~~~=~~~~& -\frac{1}{4} F_{\mu\nu}F^{\mu\nu}-\frac{\chi^2}{2\xi}\cr 
~\stackrel{\text{int. by parts}}{\sim}&~\frac{1}{2}A_{\mu}\Box A^{\mu}, \end{align}\tag{1}$$
with Lorenz gauge condition $\chi=d_{\mu}A^{\mu}$ in Feynman gauge $\xi=1$.


*We next add sources
$$ {\cal L}~=~{\cal L}_0 + A_{\mu}J^{\mu} \tag{2}$$
in the form of point charges
$$\begin{align} J^{\mu}(x) ~=~&\sum_i e_i \int\!d\lambda_i~\dot{x}_i^{\mu} \delta^4(x-x_i), \cr
\dot{x}_i^{\mu}~\equiv~&\frac{dx_i^{\mu}}{d\lambda_i},
\end{align} \tag{3}$$
which satisfy the continuity equation $d_{\mu} J^{\mu}=0$. Here $\lambda^i$ is a world-line (WL) parameter for the $i$th point charge.


*The EL equations are $\Box A^{\mu}\approx-J^{\mu}$, with solutions
$$\begin{align} A^{\mu}(x)~\approx~& \int \! d^4y~G_F(x-y) J^{\mu}(y)\cr
~=~&\sum_i e_i \int\!d\lambda_i~\dot{x}_i^{\mu} ~G_F(x-x_i), \end{align}\tag{4}$$
where the Feynman Greens function is
$$\begin{align} 4\pi G_F(x)~=~ &\delta(x^2)~=~\frac{1}{2r}\sum_{\pm}\delta(t\pm r), \cr
 r~=~&\sqrt{x^2+y^2+z^2}, \cr
 \Box G_F(x) ~=~&-\delta^4(x),\end{align}\tag{5}$$
cf. e.g. this Phys.SE post.
This instills a symmetry between retarded and advanced propagations, and will lead to a relativistic generalization of Newton's 3rd law, cf. Ref. 1.


*We add kinetic terms for the point charges
$$ \begin{align}S_i~=~&\int d\lambda_i ~L_i, \cr
 L_i~=~& -m_i\sqrt{-\dot{x}_i^2}. \end{align}\tag{6}$$
Note that the formulation is WL reparametrization invariant.


*The full E&M action reads
$$ S~=~\underbrace{\sum_i S_i}_{\text{matter}}+\underbrace{\int\! d^4x~{\cal L}}_{\text{fields + interactions}}.  \tag{7}$$
The EL equation for the $i$th point charge is the relativistic Newton's 2nd law with the Lorentz force
$$ \dot{p}_{i,\mu}~\approx~e_iF_{\mu\nu}(x_i)\dot{x}_i^{\nu}.\tag{8} $$


*If we integrate out the $A_{\mu}$-field in the action (7) we get the non-local direct interparticle action
of Schwarzschild, Tetrode & Fokker:
$$\begin{align} S~\stackrel{A}{\longrightarrow}~&\sum_i S_i+\frac{1}{2}\int d^4x\int d^4y  ~J_{\mu}(x) G_F(x-y)J^{\mu}(y) \cr 
~=~&\sum_i S_i+\frac{1}{2}\sum_{i\neq j} e_i e_j \int\!d\lambda_i~\int\!d\lambda_j~ \dot{x}_i\cdot \dot{x}_j~G_F(x_i-x_j) ,\end{align}\tag{9} $$
where we have discarded the singular self-interaction terms in the double-sum of eq. (9).
References:

*

*J.A. Wheeler & R.P. Feynman, Classical Electrodynamics in Terms of Direct Interparticle Action, Rev. Mod. Phys. 21 (1949) 425 (PDF).

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$^1$ We use the Minkowski sign convention $(-,+,+,+)$ (which agrees with Ref. 1) and put $c=1$.
