They hover for a second or two as the female gets an egg ready to be laid, then they dip down and drop off the egg. Here's a short video clip showing an egg-laying dip:
This video was shot at 240 frames per second, and a frame-by-frame analysis reveals a number of details of the tandem flight. First, the wingbeat frequency during this hovering flight is approximately 48 beats per second. That's a blur in real time, but in the slow-motion video you can follow each individual wing beat.
In addition, it's possible to see how the wing beats of the male and female are related to one another. One might think they flap in sync, but that's not the case. Alternatively, one might think they flap independently, but again, this is not the case. What they do is the following:
(1) The female's hindwings flap first.
(2) After 2/5 of a cycle, that is 144˚ of phase later, the female's forewings flap in unison with the
male's hindwings. This is the key aspect of the hovering flight.
(3) The male's forewings flap 1/5 of a cycle (72˚) later.
(4) Finally, 2/5 of a cycle (144˚) later the female's hindwings flap for the next cycle.
There's a definite relationship between how the male and female flap, it's just not what one might imagine.
Here's a plot that shows the wing positions for both the male and female as a function of time for two cycles:
This same type of phase relationship between the male and female has been seen in Red-veined Meadowhawks. It will be interesting to see how widespread this behavior is among other species.