Can anything from a molded house be safely moved to a new house without any further harm for health?
Can spores be wiped out from clothes and furniture?
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The CDC has a great website on cleanup after mold. Also, this is a simple pamphlet. The degree of decontamination needed depends on the item and the extent of the mold. Whether something needs to be thrown out depends on multiple factors. There is too long a list to give a full discussion here, but the fundamentals include:
Mold has known deleterious health effects, so especially if you have allergies to mold, make sure you do read the CDC recommendations if you've experienced mold problems. Spores can be transported to a new location and grow there if it encounters favorable circumstances (damp rooms, leaking roof, etc).
Note: this is my personal experience, not a scientific opinion or anything like that. I hope it helps.
I had a similar situation to you three years ago. We moved out of the damp apartment, full of black mould. We cleaned all of our furniture with ordinary upholstery shampoo, washed our clothes with normal detergent, etc.
We then moved to a nice, dry apartment. No doubt some spores remained on the clothes, etc. However, they did not cause any health effects that we know of. We have moved several times since then and have never had symptoms of anything. The spores need moisture to grow into mould, or they'll die.