I'll start with the French army, which is what I know best. French troopers of horse initially wore crossbelts until the early 1680's as shown in the illustration below reproduced by René Chartrand in his Osprey MAA on the army of Louis XIV :
On December 10 1683, Louvois ordered a change in the accoutrements of the cavalry. From then on, the sword would no longer be supported by a shoulder belt, but by a waistbelt (A) while the carbine remained slung from a bandoleer (L). The dragoons also abandoned the crossbelt arrangement, using only a waistbelt from which the bayonet was also suspended (H) (the plug bayonet shown here would have been replaced with a socket bayonet by the early 1700's)
The drawings above are taken from Surirey de Saint Remy's
Mémoires d'artillerie (published 1697 and dealing with more than just artillery equipment). Here's what the waistbelt and bandolier looked like on cavalry troopers in this 1695 engraving by Nicolas Guérard :
The waistbelt was supposed to be buckled over the coat, though Guérard occasionally shows it worn under the open coat - possibly during the summer or in hot weather :
Since this is a camp scene, note that not all troopers are wearing the carbine shoulder belt.
One morning I shot an elephant in my pajamas. How he got into my pajamas I'll never know.