I hope you are doing fine.
Which version of LR is it that you are talking about ?
By the way for any machine that has a better hardware, technically performance should increase.
Oh sorry - I'm using the latest Ligthroom CC Version.
Exactly what parts and what actions in Lightroom are slow on your old computer?
Browsing the pictures for escample (especially in devellop mode) could be faster. and if i have to take captions/cut lots of picture frames the computer can get horrible slow ...
i think you're always watching out for better performance as a photographer. I usally deal with photo-projects of 1000 to 4000 pictures and have to make selections and get them ready in about 10-30 hours of work. so every little interruption, espacially when they occur on every photo I deal with, can mean sitting lots of more hours at every photo-project.
Do you perform a lot of spot healing or brushing in Lightroom?
Sometimes yes. But mainly brushing and gradients...
Lots of brushing and spot healing can cause LR to slow down as you are experiencing, and as far as I know, changing your hardware to the most recent probably won't help much. Nevertheless, if you're going to upgrade your hardware, you'd want to get the fastest CPU you can afford to have the biggest impact on this problem.
A better solution, as far as speed is concerned, is to do the brushing and spot healing in Photoshop or Photoshop Elements and not do this in Lightroom.
One other thing you can try in Lightroom is to perform lens corrections only after you have finished brushing and spot healing.