Headaches during or after EMDR are, in my experience, nearly always related to latent dissociative conditions. The EMDR pulls forward a previously hidden conflict, whether its the front part of self that doesn't want to "own" another state that the EMDR is pulling forward, or the back part of self that doesn't want to be pulled forward or that doesn't want a secret to come out, etc.

I wouldn't recommend doing EMDR on a dissociative client without having done the preliminaries - the stabilization and containment, but also the ego state preparatory work. Then if a headache comes up during EMDR one can deal with it on the spot, but its less likely to come up.

In my opinion (no data, just my clinical impression), tapping or sound BLS its more possible for a latent ego state to "hide" during the EMDR than with the eye movements. With eye movements you can see when the eyes stop tracking because there is a dissociated part at arms length. That's what's possibly happening when the headache occurs the next day.

Generally, I'd interview "the part behind the headache the next day," using ego state therapy, asking it to come into the conference room "so we can understand your concerns."