A question: Do you mean that you want to upgrade the OLDER machine from 32-bits to 64-bits? You miswrote "I want to install 8.1 32 bit on my older laptop," but that's the way it already is. If this is the case, what make and model is the old one? Some machines simply don't have 64-bit processors and won't run a 64-bit OS.
"...using my newer laptop." You can't transfer the OS from one laptop to the other. You could certainly do a clean install of windows 64-bit on the32-bit machine if you wanted to. The main risk is failing activation. You should also have a complete backup of your drive so that, if the process fails for any reason, you can restore to where you were with 32-bit. You do know that the process will destroy any data (pictures, documents, anything) that you have on the machine, do you not? You will need a windows distribution disc or downloaded ISO.
From a valid source only, please. Note that discussions of torrenting will be removed.