I am not a practitioner, but I believe the thinking goes something like this: eye movements influence further memory "consilodation" in a way that de-consolidates, and disrupts existing emotional associations. Supposedly, it is almost a direct neural pathway disruption.

Not critical at all. I did EMDR as a client and could not tolerate the eye movement. It made me dizzy. My therapist use a tool with two electrical devicesthat gently vibrate, held in each, hand for bilateral stimulation. People can also do a tap themselvs. They key is the bilateral stimulation.

Eye movement is not critical in EMDR. Any bilateral stimulation (hand taps, hand buzzers, alternating sounds in each ear, etc) will provide the same processing. The most important factor is that it is bilateral and that the client can tolerate it.

You may be interested in the following report: van den Hout, M. A., Engelhard, I. M., Rijkeboer, M. M., Koekebakker, J., Hornsveld, H., Leer, A., et al. (2011). EMDR: Eye movements superior to beeps in taxing working memory and reducing vividness of recollections. Behaviour Research and Thereapy, 49, 92–98

You might find this article interesting:

Dunn, T. M., M. Schwartz, et al. (1996). "Measuring effectiveness of eye movement desensitization and reprocessing (EMDR) in non-clinical anxiety: A multi-subject, yoked-control design." Journal of Behavior Therapy and Experimental Psychiatry 27(3): 231-239.

Abstract: Twenty-eight subjects from a university's subject pool were paired on sex, age, severity, and type of stressful or traumatic incident. One subject in each pair was selected to receive EMDR; the experimental partner spent the same amount of time receiving a visual (non-movement) placebo. Subjective units of discomfort (SUD) scores and physiological measurements were taken prior to and following treatment. Analysis of physiological measurements and self-reported levels of stress were performed within and between each group. While the EMDR group showed significant reductions of stress, EMDR was no better than a placebo. This suggests EMDR's specific intervention involving eye movement may not be a necessary component of the treatment protocol.

As others have said it is the bilateral stimulation (BLS) that is the critical feature. Whether eye movements are better than other forms of BLS remains unknown.

Some studies have started to look at this but the numbers of participants involved are low and they are often done using non-clinical populations.

A definitive answer may not be possible until we can understand the neurological mechanisms behind the BLS and memory retrieval and reconsolidation and we are not close to that yet.

Eye movement is not a critical component of EMDR. You can use any bilateral stimuli, but , maybe the most important on that part of the process is dual attention.