I have an 8 year old that is constantly in motion too. He was first diagnosed with sensory processing disorder (sensory-seeking) before ASD. A weighted vest helped him plus OT sessions. Sometimes just having him jump up and down a bunch of times helps to get him the sensory input he needs and he calms down; we make a game out of it for fun.

My son jumps and climbs on everything. Sofa chairs high tables what ever he sees. It is the sensory he gets from it. Sometimes when he gets in his moments I will play with him holding his hands letting him jump to make sure he safe then tickle him and he usually gets tired for a moment to rest a bit. But not for long. I found this is a great time to teach my son things like you want to jump then jump to letters I call out.

One thing I hate is when I take him to other peoples home and he wants to jump on their items now that is the hard one lol.

My son does similar and really enjoys the sensation of falling. OT has identified that it is the need for instant deep pressure. We are currently working on ways to satisfy this need safely so if we come up with anything successful I will let you know. To date everything other than the trampoline has been a big fat fail and I cannot bring a 16ft trampoline in the house

My son is 15 and started this about 5 months ago. I have tried so many replacement behaviors-trampoline, push ups, jumping jacks, compression. But he still prefers jumping into drawers, climbing shelf and on top on counters. I just started an ADHD medication. Have you had any luck with anything?

Thank you for your reply Wse. I have never thought about things like rock climbing or bungee jumping and I'm very curious how he would do! I am going to look into our local gyms and see if they have options for my child. It's actually an amazing talent that he has. All he needs is a 1 inch ledge or 2 walls within 2 feet of each other and he's 10 feet in the air in no time! Although I must say I get very upset, angry and discouraged when I seek advice from people who don't have children with autism. They make lecturing remarks like, "You need to work harder to teach him not to climb," and "You're not watching him closely enough." and "if you don't put the time in now, wait until he's bigger." ... My life is devoted to him and such accusations are hard to hear. I have worked with him for 5 years with many specialists using many different approaches. But if you could see the intense need and drive in him when you try to redirect him from things he's climbing- it really is like taking a cigarette from a chain smoker. His teacher sees the same thing.
So, again I thank you Wse for the suggestions. I am going to call local gyms. His teacher keeps telling me, you can't just stop a behavior from happening. You have to replace it with a more appropriate one.