Is it normal for the woofer to move in/out slowly

My only tubed piece is the pre-amp. It is the only piece I turn off when not in use. I noticed that when I turn it off the woofers on my speakers move in and out rather slowly while I hear what I assume are the tubes discharging.

Is that normal and more importantly can it cause damage to the speakers?

IMO it is asking for trouble turning off a pre amp while the amps are on. Why do you turn it off? Maybe I am old school, but I have always left my pre amps on.
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Well, some Tube preamp manufactures recommend them turned on all the time, I've had mine on for over 2 years with original tubes...
1. It's not abnormal

2. It's only bad if it drives the woofers to their mechanical limits.

If you really wanted you could hang a scope on your speaker terminals and see what's happening.

1-1.5 volts peak is fine even for tweeters (That's what I get in the power amp connected directly to my main actively tri-amplified system's tweeters and they have over a thousand power cycles on them).
Bokfudo, My speakers does this. I have a tube pre which I turn off after listening, have done this for years with no issues. Back to your question, yes my speakers does this and I don't see it as a problem hence have not investigated. But I will not worry about it.