Is it support of or support to?

I would say that "support to" is used to mean a literal support such as to hold up a roof. He added a steel brace to give more support to the roof. I would say "support for" to indicate approval for...

You can use either of the terms:Logistics support or logistical support

Give moral or psychological courage..

Technically child support is for ALL the needs for the child, except medical.

reinforcement (a military operation (often involving new supplies of men and materiel) to strengthen a military force or aid in the performance of its mission)