X BREED characteristics?

No, you can follow no assumption when crossing two breeds - especially when the gene pools are phytogenically far removed (ie, measured as independent development over time).

For instance, Aussie and Border Collies are pretty close to each other.  But, I have a BC/Aussie mix here who looks and works like a purebred BC - a real hardcore Aussie person might go "hmmmmm" at him but walk away with no questions answered for sure.  He acts 100% Aussie.  Well, maybe 90% Aussie, but really his entire personality is well within the range of Aussie temperaments.  It's made him rather hard to place.  BC people will talk to me because he looks like a blue merle BC (way cool) but when I say, "Yes, but are you ready for X-Y-Z which is not a bit like your typical BC?" they are like, "Um, don't call me, I'll call you . . ."  Like wise I can't really even get a foot in the door to Aussie people, 90% of which want a nice pet, not the dog who is pictured doing fairly advanced livestock work, and looks like a BC complete with full tail which he holds down (no "Wiggle Butt";).

I've fostered and known a few Pyrenees/sheepdog mixes (either collies or Border Collies or Aussies), and every one of them tends to take after the Pyrenees, though they end up looking pretty much like giant, very hairy, white factored, whatevers. 

I knew one Beagle/Aussie mix that was bigger than either breed should have been (eighty pounds!), but looked like a purebred Aussie otherwise, though with more white and coloring that you would have only realized was beagle colors, if someone said, "Beagle."  But his personality was 100% Beagle - it was hilarious!  He was a wonderful dog - he was born to a breeder of both show Beagles and top show Aussies, and his parents were both large for their breed.  Apparently all the puppies were just like him, too, only some were more Aussie-ish, and some were smooth caoted.  Every one of them went aaaaaaa-ROOOOOOOOO! though and had the happy wavy tail thing.  Only in a 60 to 80 pound plume tailed Aussie thing, well, no one with those puppies kept breakables at tail level! 

I would have to say yes and no. It isn't really possible to tell. There are many genes governing the dog, so the genes that produce the phenotype of one breed might be inherited but not the genetics for the temperament or personality traits. Some might look and act more like one breed and others might look like one but act like the other, some might be a mixture of both temperament even if they look like one or the other.

 Thanks for your replies. So if I see a BeagleX for sale or adoption and to my eye it looks like a lot of beagle in the mix, I should have no expectation that this particular dog would be any more likely to have the temperament of a beagle, than the dog next to him described as a poodle mix?