Fill the data of an Association based on a list

Another way (assuming that the keys are ordered)

AssociationThread[list1 -> Catenate[MapThread[ConstantArray,
         {Values[asc], Differences[Prepend[Keys[asc], 0]]}]]]

<|1 -> a, 2 -> b, 3 -> b, 4 -> b, 5 -> c, 6 -> c, 7 -> c, 8 -> d, 9 -> d, 10 -> d|>

Piecewise does what you need and here is a readable way to use it:

AssociationMap[
  Evaluate[
    Piecewise[KeyValueMap[Function[{k, v}, {v, # <= k}], asc]]
  ] &
, list1
]

Table[i -> asc[NestWhile[# + 1 &, i, ! MemberQ[Keys[asc], #] &]], {i,list1}] // Association
(* <|1 -> a, 2 -> b, 3 -> b, 4 -> b, 5 -> c, 6 -> c, 7 -> c, 8 -> d, 9 -> d, 10 -> d|> *)