Why is 丢的东西 placed in the middle of 孩子们丢的东西都找到了?
孩子们丢的东西都找到了



The object is "东西"
东西都(被)找到了 - things have all been found (被 omitted)
孩子们丢的 is an adjective phrase that describes the object 


"things? what things?" 

"Things that was lost by the kids" (孩子们丢的)

you can use 的 to turn any sentence into an adjective phrase

Example:

美国进口 - American import

美国进口的 - that is American imported

美国进口的牛肉 - beef that is American imported