Do the photons emitted along with the gravitational waves take more time to reach earth than the ones emitted after them? I'm puzzled because the gravitational waves do warp the spacetime locally. They do it continuously as they propagate. So the photons travelling in tandem with these waves must be constantly following a curved path and hence must be travelling more distance to reach earth than 'normal' photons from the same source that travel without these disturbances. So do they take more time to reach earth?
 A: In vacuum, that is, in absence of matter and electromagnetic interactions, photons and gravity waves follow the same geodesics (if gravitons are actually massless... Let us assume it for we have no strong evidences for the contrary). In their propagation photons are not affected by the ripples and the stretches in space-time caused by the waves, because they are travelling along with them. There are no front crossings.
You can figure this out looking at the photons as a surfer riding a wave; if he moves along with the wave, that is, he's at rest with respect to the wave, and he doesn't perform any trick or strange path, the space he covers is exactly the same as if the sea were calm. He doesn't measure the slope and the curves in the surface of the water, which would constitute and increment in the path length respect to when the sea is calm.
In conclusion, neglecting the electromagnetic interaction of the photons with the interstellar medium, photons travelling in tandem with the gravity waves, and those leaving later from the same source, take exactly the same time to reach us.
EDIT: consider also that the stretching effects of gravitational waves are only transverse (at least in General Relativity) to the propagation direction of the waves. Thus, a light ray travelling in their same direction, won't be effected by any effect of dilatation/compression of the space-time. This is indeed the working principle of laser interferometric detectors: measure the interference of two orthogonal light beams.

A: You are right - it would take longer for light (and gravity and anything else) to make the trip during the event than after. But the effect is tiny, as gravitational waves shrink and expand space in virtually equal amounts - and in the linear regime - which the waves are in for virtually the whole trip, the compression and expansion is perpendicular to the direction of motion. 
It is only because of the non - linear behaviour of general relativity that there is a very slight overall increase in time. I don't know what the amount of time would be, but something like 99% of the extra delay would occur in say the first million km from the event. So even if the effect was say 1% over the first million km, that would amount to only an extra (light takes 3 seconds to go a million km) 0.03 seconds. Then virtually nothing for the next billion years of travel. Those numbers are for illustration only - I'm sure someone has done the calculation somewhere. 
After a million km of travel, the gravitational waves are well into the linear regime and the effect turns from a small effect into a completely ignorable tiny amount. 
So 1.3 billion years for light travel without the event, and 1.3billion years + 0.03 seconds for light travelling in the event.  
Please note that the 0.03 seconds is nothing more than a complete guess, I would like to see the real number.
Gravitational waves will only look like the neat linear fully transverse waves as one gets out of the intensive emission region. 
