How to apply HT gradually in a valve amplifier?
Since it takes the tubes some time to warm up, what you need is a time delay relay that starts to time out when mains power is applied to the amplifier and then closes a set of contacts between the B+ and its load(s) when the proper time has elapsed, perhaps 30 seconds to a minute, depending on which of the tubes takes the longest time to warm up.

Then, when mains power is cut off some time later, the relay opens immediately, allowing the tubes to cool down with no stress from the B+.

The relays are available commercially - quite pricey - and are called ON-DELAY relays, or you could easily put one together if you're so inclined.