How to understand the Einstein delay? In pulsar timing, one of the post-Keplerian parameters is the Einstein delay $\Delta E$. The total `flight time' of a photon emitted from a pulsar is then given by
$$ \Delta t = \Delta R + \Delta E + \Delta S$$
where $\Delta R$ and $\Delta S$ are the Roemer and Shapiro Delays.
The Einstein delay is apparently due to both Doppler shift and gravitational time delay.
I understand the gravitational time delay part, but cannot see how the motion of the pulsar and consequent Doppler shift would affect the total photon flight time?
 A: In pulsar timing you can't measure the absolute travel time from the pulsar to Earth.  Instead you measure changes in the time of arrival relative to the expectation based on a model.  You can think of it as changes in the time between pulses.
The Einstein delay covers changes in time of arrival due to the gravitational field of the pulsar and time dilation from its motion.  Because the pulsar is moving relative to the Earth, its intrinsic rotational frequency is changed due to time dilation.  In pulsar timing people will use Doppler shift to refer to a shift in the rotational frequency.  Since we get one pulse per rotation the frequency of pulse arrival is Doppler shifted.  (This is different than the frequency of the radio waves shifting, which also occurs.)
Assuming the pulses originate at the same location in the pulsar's atmosphere, then they all experience the same gravitational delay.  If the pulsar is in a circular orbit, then it moves at a constant speed with a constant acceleration.  This means all of the pulses will have the same motional time delay.
Since all pulses are delayed by the same amount, it is impossible to measure a relative change.  That's why the Einstein delay part of $\Delta t$ depends on eccentricity, $e$.  In an eccentric orbit the speed and acceleration of the pulsar change.  Near periastron the pulsar moves faster and the time dilation is larger.
The post-Keplerian parameter $\gamma$ is the amplitude of the Einstein delay.
$$\gamma = e\, \left( \frac{P_b}{2\pi}\right)^{1/3}\,\left(\frac{G}{c^3}\right)^{2/3}\,\frac{m_c (m_p + 2m_c)}{(m_p+m_c)^{4/3}}$$
That's the maximum Einstein delay experienced during an orbit.  The actual delay oscillates as a function of orbital phase.  By fitting the amplitude of multiple post-Keplerian effects the mass of the pulsar $m_p$ and its binary companion $m_c$.
A: It's just the special relativisitic part of the time dilation.
In special relativity an observer sees a time dilation effect on objects moving relative to them near the speed of light.
In General relativity there is also time dilation due to gravitational effects.
Only Time Dilation due to relative velocity is accounted for by Special Relativity. Gravitational Time Dilation is another effect. Both effects are included in the formulation of General Relativity.
The separation of the effects into aproximately addative terms is possible when $\large\frac{GM}{R c^2}≪1$. However in your scenario this may not be the case, which remains a point of confusion.
Doppler shift, as you point out, impacts frequency and not velocity.
