diff --git a/mypy/subtypes.py b/mypy/subtypes.py
index 7e37751b1c15..4fd3f8ff98ca 100644
--- a/mypy/subtypes.py
+++ b/mypy/subtypes.py
@@ -1651,7 +1651,12 @@ def _incompatible(left_arg: FormalArgument | None, right_arg: FormalArgument | N
                 continue
             return False
         if not are_args_compatible(
-            left_arg, right_arg, ignore_pos_arg_names, allow_partial_overlap, is_compat
+            left_arg,
+            right_arg,
+            is_compat,
+            ignore_pos_arg_names=ignore_pos_arg_names,
+            allow_partial_overlap=allow_partial_overlap,
+            allow_imprecise_kinds=right.imprecise_arg_kinds,
         ):
             return False
 
@@ -1676,9 +1681,9 @@ def _incompatible(left_arg: FormalArgument | None, right_arg: FormalArgument | N
             if not are_args_compatible(
                 left_by_position,
                 right_by_position,
-                ignore_pos_arg_names,
-                allow_partial_overlap,
                 is_compat,
+                ignore_pos_arg_names=ignore_pos_arg_names,
+                allow_partial_overlap=allow_partial_overlap,
             ):
                 return False
             i += 1
@@ -1711,7 +1716,11 @@ def _incompatible(left_arg: FormalArgument | None, right_arg: FormalArgument | N
                 continue
 
             if not are_args_compatible(
-                left_by_name, right_by_name, ignore_pos_arg_names, allow_partial_overlap, is_compat
+                left_by_name,
+                right_by_name,
+                is_compat,
+                ignore_pos_arg_names=ignore_pos_arg_names,
+                allow_partial_overlap=allow_partial_overlap,
             ):
                 return False
 
@@ -1735,6 +1744,7 @@ def _incompatible(left_arg: FormalArgument | None, right_arg: FormalArgument | N
             and right_by_name != right_by_pos
             and (right_by_pos.required or right_by_name.required)
             and strict_concatenate_check
+            and not right.imprecise_arg_kinds
         ):
             return False
 
@@ -1749,9 +1759,11 @@ def _incompatible(left_arg: FormalArgument | None, right_arg: FormalArgument | N
 def are_args_compatible(
     left: FormalArgument,
     right: FormalArgument,
+    is_compat: Callable[[Type, Type], bool],
+    *,
     ignore_pos_arg_names: bool,
     allow_partial_overlap: bool,
-    is_compat: Callable[[Type, Type], bool],
+    allow_imprecise_kinds: bool = False,
 ) -> bool:
     if left.required and right.required:
         # If both arguments are required allow_partial_overlap has no effect.
@@ -1779,7 +1791,7 @@ def is_different(left_item: object | None, right_item: object | None) -> bool:
             return False
 
     # If right is at a specific position, left must have the same:
-    if is_different(left.pos, right.pos):
+    if is_different(left.pos, right.pos) and not allow_imprecise_kinds:
         return False
 
     # If right's argument is optional, left's must also be
diff --git a/test-data/unit/check-parameter-specification.test b/test-data/unit/check-parameter-specification.test
index eb6fbf07f045..d65a0214b599 100644
--- a/test-data/unit/check-parameter-specification.test
+++ b/test-data/unit/check-parameter-specification.test
@@ -1687,9 +1687,18 @@ P = ParamSpec("P")
 T = TypeVar("T")
 
 def apply(fn: Callable[P, T], *args: P.args, **kwargs: P.kwargs) -> None: ...
+
 def test(x: int) -> int: ...
 apply(apply, test, x=42)  # OK
 apply(apply, test, 42)  # Also OK (but requires some special casing)
+apply(apply, test, "bad")  # E: Argument 1 to "apply" has incompatible type "Callable[[Callable[P, T], **P], None]"; expected "Callable[[Callable[[int], int], str], None]"
+
+def test2(x: int, y: str) -> None: ...
+apply(apply, test2, 42, "yes")
+apply(apply, test2, "no", 42)  # E: Argument 1 to "apply" has incompatible type "Callable[[Callable[P, T], **P], None]"; expected "Callable[[Callable[[int, str], None], str, int], None]"
+apply(apply, test2, x=42, y="yes")
+apply(apply, test2, y="yes", x=42)
+apply(apply, test2, y=42, x="no")  # E: Argument 1 to "apply" has incompatible type "Callable[[Callable[P, T], **P], None]"; expected "Callable[[Callable[[int, str], None], int, str], None]"
 [builtins fixtures/paramspec.pyi]
 
 [case testParamSpecApplyPosVsNamedOptional]
@@ -2087,6 +2096,52 @@ reveal_type(d(b, f1))  # E: Cannot infer type argument 1 of "d" \
 reveal_type(d(b, f2))  # N: Revealed type is "def (builtins.int)"
 [builtins fixtures/paramspec.pyi]
 
+[case testParamSpecGenericWithNamedArg1]
+from typing import Callable, TypeVar
+from typing_extensions import ParamSpec
+
+R = TypeVar("R")
+P = ParamSpec("P")
+
+def run(func: Callable[[], R], *args: object, backend: str = "asyncio") -> R: ...
+class Result: ...
+def run_portal() -> Result: ...
+def submit(func: Callable[P, R], /, *args: P.args, **kwargs: P.kwargs) -> R: ...
+
+reveal_type(submit(  # N: Revealed type is "__main__.Result"
+    run,
+    run_portal,
+    backend="asyncio",
+))
+submit(
+    run,  # E: Argument 1 to "submit" has incompatible type "Callable[[Callable[[], R], VarArg(object), DefaultNamedArg(str, 'backend')], R]"; expected "Callable[[Callable[[], Result], int], Result]"
+    run_portal,
+    backend=int(),
+)
+[builtins fixtures/paramspec.pyi]
+
+[case testParamSpecGenericWithNamedArg2]
+from typing import Callable, TypeVar, Type
+from typing_extensions import ParamSpec
+
+P= ParamSpec("P")
+T = TypeVar("T")
+
+def smoke_testable(*args: P.args, **kwargs: P.kwargs) -> Callable[[Callable[P, T]], Type[T]]:
+    ...
+
+@smoke_testable(name="bob", size=512, flt=0.5)
+class SomeClass:
+    def __init__(self, size: int, name: str, flt: float) -> None:
+        pass
+
+# Error message is confusing, but this is a known issue, see #4530.
+@smoke_testable(name=42, size="bad", flt=0.5)  # E: Argument 1 has incompatible type "Type[OtherClass]"; expected "Callable[[int, str, float], OtherClass]"
+class OtherClass:
+    def __init__(self, size: int, name: str, flt: float) -> None:
+        pass
+[builtins fixtures/paramspec.pyi]
+
 [case testInferenceAgainstGenericCallableUnionParamSpec]
 from typing import Callable, TypeVar, List, Union
 from typing_extensions import ParamSpec