For an affine algebraic group $G$ it's often convenient (and harmless)  to work concretely over an algebraically
closed field of definition $k$ while identifying $G$ with its group of rational points over $k$.
Once in a while, however, results obtained over $k$ need to be compared with results over
a bigger algebraically closed field $K$. 
    
For example, in his 1976 *Inventiones* <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002092506">paper</a>  completing the proof that a semisimple algebraic group
always has a finite number of unipotent conjugacy classes, Lusztig observed that in characteristic
$p>0$ it suffices to assume that $k$ is an algebraic closure of the prime field.   This in turn
allowed him to apply indirectly the Deligne-Lusztig construction of characters for subgroups of
$G$ over finite subfields of $k$.    Here the number of unipotent classes is moreover the same for any algebraically closed field.   To justify his reduction, he cites a "simple argument" shown to
him by Deligne (which he later told me he viewed in retrospect as "obvious").    

Independently, a formal statement of the principle was written down and proved as Proposition 1.1  in a 1997 *Journal of Algebra* paper:

[MR1474171](http://www.ams.org/mathscinet-getitem?mr=1474171) (98j:20058),

Guralnick, Robert M. (1-SCA); Liebeck, MartinW. (4-LNDIC);
Macpherson, Dugald (4-LEED); Seitz, Gary M. (1-OR),

[Modules for algebraic groups with finitely many orbits on subspaces](http://www.sciencedirect.com/science/article/pii/S002186939797068X).

J. Algebra 196 (1997), no. 1, 211–250.

Here the proof is fairly elementary, but requires more than a journal page to write down and involves an induction step left to the reader.     

> Is there a short and transparent proof (perhaps from the scheme viewpoint) that finiteness of the number of orbits of a semisimple group acting on an affine variety is the same when an algebraically closed field of definition is extended to another such field, while the number of orbits is unchanged?

It would also be interesting to know of other situations in which such a comparison occurs.
(Historical remark: In the older version of foundations for algebraic geometry developed
by Weil and others it was standard procedure to work over a "universal domain" having infinite
transcendence degree over its prime field, to permit for instance the use of "generic points".   Then it was usually troublesome to descend to a countable field.)  

ADDED: To be more precise about "the number of orbits is unchanged", implicit in Lusztig's work and explicit in the 1997 paper cited is the natural requirement on such a bijection that each orbit over $K$ should contain a point over $k$.    (It's hard to visualize a proof that gives a bijection without this refinement.)  On the other hand,  it's unclear to me whether special assumptions on $G$ such as "reductive" and "connected" are essential for the proof of a general comparison principle.