If you divest US bonds, you would probably put them into bonds from other nations (and corporate bonds from non-US companies), easiest thing is to try to find a index to track; Vanguard's BNDX tracks the Bloomberg Global Aggregate ex-USD Float Adjusted RIC Capped Index (Hedged).

In a mark to market world, the value of a bond is its acquisition cost, so buying bonds enough to raise prices increases their value, but not their coupons or their face value. It's hard to make sense of the value of a sequence of payments, it's reasonable to consider the present value and the market price is an easily justified present value for a bond.

Selling bonds and buying stocks is a different thing altogether. Selling US stocks and buying EU stocks wouldn't change the value of the underlying assets, however, having an increased stock price does have benefits for the company when issuing new shares or bonds.