Is it normal to go out of tune with bends?

It's not normal, but it does depend to a certain degree how far you are bending the string.
But it is not normal for a guitar to lose pitch are a few simple bends.
Things to check for:

Are you winding on the strings correctly? don't wind the strings around the tuning posts 20 times, this will cause the string to slip under tension.
If you know how to correctly re-string a guitar then try these other tips:

When you change strings use some 'Nut Sauce' (a product available to help the string from catching on the nut and saddle ) or you could try Graphite from a pencil and ordinary 'B' or '4B' type, just scratch the pencil over the nut a few times til you see the graphite dust laying in the slot and on the saddles.

Is it all strings losing pitch after bends?

Another thing to try try is putting a capo nice and tight on the 2nd fret, then try your bends and see if it goes out of tune, if it stays in tune and you have tried the above tips, then it could be a slipping machine head, replacements are available from allparts.com or .co.uk

Does your guitar have a trem system? if so which type?

Well thats some tips to be getting on with,

Let us know how you get on,

Good luck.

I can only go by experience, i have over 20 guitars at the moment, they range from Crafter to Gibson, my oldest guitar is a Tokai Reborn LP made in 1979, my newest is a 2005 Gibson Hummingbird, i have Tokai, Rickenbacker, Gibson, Fender, Ibanez and none of them go out of tune after a few string bends, new string are obviously a bit temperamental for a couple of hours, but they soon settle in. I find if i restring correctly i don't get strings slipping.
And of course if you spend £49.95 on a new guitar then the tuners may well be rubbish, but im assuming he has a half decent guitar.
And i still stand by my statement, it is NOT normal for a guitar to go out of tune after a few string bends.
I gig mainly with 3 guitars, a Gibson SG standard, a Gibson LP standard, and a 2004 Jap made Tokai Love rock, i play them for 2-3 hours in a gig, and rarely retune them, definatly not during a set, i have a little check during the break and they rarely need tuning, i play solo's all night, plenty of string bends in hot humid conditions. I restring correctly, and use graphite on the nut and bridge.