The breakdown and reassembly of proteins is a ubiquitous process within cells, and yes this is expensive but transport is expensive, too, and recycling has the added benefit of dealing with proteins that become misfolded or otherwise damaged as well as allowing for transcription and translation to regulate overall protein levels. The Toyoma & Hetzer 2013 review (see references below) cites estimates of the median protein half-life in a non-dividing mammalian cell to be about 43 hours, though they discuss some exceptions that do last much longer.
However, it is possible to relocate transmembrane proteins via vesicles.
Essentially the same way proteins are put in the plasma membrane in the first place, but in reverse. Endocytosis is often taught as a way that cells (often macrophages are used as an example) take in materials from the outside world, but it is just as applicable to internalizing chunks of membrane. Nature has an entire 'web focus' on entocytosis reviews.
One situation where this happens is in synaptic plasticity; receptors can be internalized or trafficked to the membrane to decrease or increase the potency of that synapse, respectively (see Carroll et al., 2001), but similar processes occur all over (and some proteins constantly cycle back and forth; see Trowbridge et al. 1993 for a more general review, though a bit dated).
However, to your specific question, I don't know specifically of examples where transmembrane proteins are literally cycled from one side of the cytosol to the other. There may be examples I am not aware of, but my understanding is that it is more typical to think of a "store" of certain proteins in endosomes, from which they can be trafficked to the membrane and reinserted via exocytosis. Probabilistically, some will be transported from one place to another, but not so much in a stepwise fashion.
Toyama, B. H., & Hetzer, M. W. (2013). Protein homeostasis: live long, won't prosper. Nature Reviews Molecular Cell Biology, 14(1), 55.
Carroll, R. C., Beattie, E. C., von Zastrow, M., & Malenka, R. C. (2001). Role of AMPA receptor endocytosis in synaptic plasticity. Nature Reviews Neuroscience, 2(5), 315.
Trowbridge, I. S., Collawn, J. F., & Hopkins, C. R. (1993). Signal-dependent membrane protein trafficking in the endocytic pathway. Annual review of cell biology, 9(1), 129-161.