Hi Paul
We moved here last year, my wife is Slovakian and we have a e year old son.
From what I know, if you are unemployed for at least 6 weeks in the UK you can then apply for that benefit to be transferred over to Slovakia for 3 months. We didn't get it because the lady in job centre said she can email me the forms so we left, and once in Slovakia she said we have to pick them up in person so at the end we left it, couldnt be asked to deal with all the bureocracy. more info here
You won't get any unemployment benefits in Slovakia because you have never worked here. You can apply for child benefits, 30eur a month per child. And if you have small kids and your wife is planing to stay home with them you can apply for parental benefit 200 eur per month per child up to 3 years of age. Do you mind me asking what's bringing you to Slovakia? I love it here, wanted to come ages ago but everybody was saying R u crazy? Slovakia? Which part of SVK are you moving to?
Hope I helped a little