I have used Valspar primer to cover semi-gloss trim paint & found it to work pretty well. I only had trouble in 2 small spots and repainted with the primer & it took. I suspect we may not have washed those areas well. I did wash down the trim before painting with the primer.
Answer for Painting semi-gloss trim and priming.
Many manufactures make the claim "Will stick to glossy surfaces without sanding". The only primers that I use in this situation are from Zinsser or XIM. Zinsser's Cover Stain and 123 acrylic are both very good and XIM 400 series primers will stick very well. Both companies have primers that are tintable.
I never used the Valspar product so I can't say if it will work or not.