Welcome to the friendliest forum on the web. Your question is a good one and in particular due to the expense of ammo and components today.
This is my take on it. I am not a gunsmith but have owned and operated an indoor range with a great rental program for the past twenty plus years.
There is no conversion as the 40 is a 10MM short. Same diameter bullets. Bullet weights may vary but that will be your choice when buying the rounds you prefer to shoot.
I have tried 40 in several 10MM guns and found only that some may not eject the spent cartridges reliably. For instance, I have fired 40 through the XDs and the Glocks on many occasions and found the 40s will cycle "most" of the time. The key word being "most", not always.
I have never replaced the main-springs to see if that would clear the problem, but as far as the other pieces and parts of the gun functioning, the rounds fit and feed out of the mags and fire as acaurately as though I were shooting the 10MM.
I do not have a 10MM so am not forced to try the change in a lighter main recoil spring to see if it would make the difference.
If your 10MM is going to be your defense gun or if it even has the probability of having to be used as a defense tool, then I would bite the bullet and either feed it the ammo it was designed for or trade it for a 40 if that is your choice of caliber for the future.
If it is to be a target gun then go for it and cycle the rounds manually when they do not eject. Just a suggestion, try some factory 150 to 165 grain 40s through your gun to see how it likes them before you go to a lot of expense to make changes.
A good, firm grip will make a huge difference in the consistency of the 40s cycling the gun. A weak or limp wrist will make it fail unequivocally.
There is no danger that I am aware of in trying or shooting the 40 S&W in the 10MM guns.
Good luck and I hope this helps a little.
UF