Can you give examples for a halide ion reacting with interhalogen compound to give polyhalides?
TLDR: There is no exact generalized reaction. The reaction depends on type of interhalogen compound(value of n) and the nature of metal halides(ionic or covalent)




$\ce{XX^{'}}$ form


Let us take iodine monochloride($\ce{ICl}$). If it is reacted with ionic chloride like $\ce{KCl}$, it will form $\ce{ICl2-}$ ion.

$$\ce{MCl + ICl -&gt; M+ICl2-, (M=K, Rb)}$$

If a different counter-ion is taken like bromide, it will form $\ce{ClBrI-}$ ion

$$\ce{ICl + KBr -&gt; K+[ClBrI]-}$$

But if we add pure diatomic halogen, it will form a additive product

$$\ce{Cl2(excess) + ICl -&gt; ICl3}$$

Covalent halides like $\ce{AlCl3}$ however gives different product:

$$\ce{ICl + AlCl3 -&gt; I+ + AlCl4-}$$


$\ce{XX^{'}3}$ form


Let us take iodine trichloride. It will form $\ce{ICl4-}$ ions on reacting with ionic chloride:

$$\ce{MCl + ICl3  -&gt; M+ICl4-}$$

If we take any other counter ion like fluorine, it will form $\ce{MICl3F}$ which is again kind of an additive product.


$\ce{XX^{'}5}$ form


Taking chlorine pentafluoride as examples, reacting with covalent fluorides like $\ce{AsF5}$ and $\ce{SbF5}$ will give adducts: $\ce{ClF5.AsF5}$ and $\ce{ClF5.SbF5}$ 

Iodine pentafluoride gives iodine heptafluoride on reacting with fluorine at 250 C 

So, there is exact no generalized reaction. It all depend upon type of inter-halogen compound, reaction conditions and nature of metal halide. The close generalization I can make is that interhalogen compound of form $\ce{XX^{'}_n}$ if reacted with ionic metal halide will form metal polyhalide of form $\ce{MXX^{'}_{n+1}}$ or $\ce{MXX^{'}_{n}X^{''}}$(if a different counter-ion is taken). Additive product is formed when inter-halogen compound is reacted with pure diatomic halogens provided the second halogen is big enough to occupy electrons like bromine or iodine. In some case, covalent halides forms adducts with interhalogen compounds.

Thus the generalized reactions are(of course, there are certain exceptions):

$$\ce{X^{'-} + XX^{'}_n -&gt; XX^{'-}_{n+1} (ionic)}$$

$$\ce{X^{'}_2 + XX^{'}_n -&gt; XX^{'}_{n+2}}$$

Reference


http://www.vpscience.org/materials/Study%20Material%20on%20d%20Inter-halogen%20%20Compounds%20By%20Dr.%20D.%20M.%20Patel.pdf (reaction sources)