There are a couple of schools of thought to this.
1 Weathering done will change over time and look more natural.
2 Items not weathered will look a bit stark to start with but will weather over time.
You pays your money and take your chance, I personally prefer a little subtle weathering then let nature take its course. Though if you really are concerned you could make a solution of Garden dirt with water and weather with that. Some will get into grooves or imperfections nicely and hang around, other bits will wash off in time as nature starts to do it's thing. But not undooable till that has happened.
Another good weathering option is to use Black Concrete Dye Powder mixxed up with fresh water to make a not too dark solution, this works very well on concrete but can be built up if the effect is too light. Just letvthings dry between applications.
JonD