But is the other layout based on the same table occurrence?
This is a key question. If you see the same exact name in Layout Setup | Show Records From for both layouts, they both refer to the same found set of records and you can just add these steps to the end of your script:
Set Error Capture [on]
Perform Find 
Go To Layout [//specify other layout here]
If they are not based on the same table occurrence, then you will need to make two changes:
Change to layout #2 before specifying find criteria
Change the set field steps to enter search criteria into the date field(s) of this other layout's table occurrence.
Both layouts is in the same table.
I want to get the date(s) found from either Date1 og Date2 in a new field "founddate" in layout #2. How do I do this?
How do you determine whether to refer to date1 or Date2? What logic determines from which field to copy this data?
You have also found what is potentiall an entire group of records, in which of these records will you modify the founddate field with this date?
if "founddate" is defined in the same table, all you need is one of these two steps:
Set Field [ExampleTable::founddate ; ExampleTable::Date1 ]
Set Field [ExampleTable::founddate ; ExampleTable::Date2 ]
But you have to determine from which table to get the date and into which found record to enter it.
Depending on how you use founddate, you may not need this field at all or you may find it useful to specify global storage for it.